1. Introduction
By space we will mean Tychonoff topological space, and by countable we will mean finite or countably infinite. Recall that a space is realcompact if it is homeomorphic to a closed subspace of
$\mathbb {R}^\kappa $
for some cardinal
$\kappa $
. Given a realcompact space X, we will denote by
$\mathsf {Exp}(X)$
the smallest infinite cardinal
$\kappa $
such that X is homeomorphic to a closed subspace of
$\mathbb {R}^\kappa $
. We will call
$\mathsf {Exp}(X)$
the realcompactness number of X.
It is well-known that Lindelöf spaces are realcompact (see Theorem 3.2 or [Reference Engelking3, Theorem 3.11.12]). As pointed out by van Douwen (see [Reference van Douwen2, Exercise 8.23]), for every infinite cardinal
$\kappa \leq \mathfrak {c}$
there exists a separable metrizable crowded space X such that
$\mathsf {Exp}(X)=\kappa $
.Footnote
1
However, the situation turns out to be more interesting for countable crowded spaces. More specifically, the following is our main result, which exhibits an unexpected connection between realcompactness and the pseudointersection number
$\mathfrak {p}$
.
Theorem 1.1. Given a cardinal
$\kappa $
, the following are equivalent
$:$
-
• There exists a countable crowded space X such that
$\mathsf {Exp}(X)=\kappa $
. -
•
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$
.
After the preliminaries discussed in Sections 2 and 3, we will devote the entirety of Sections 4–6 to the proof of the above theorem. In Section 7, we will give an analysis of the pseudocharacter of countable subsets of products of first-countable spaces, which will allow us to obtain the upper bound for the realcompactness number of a countable space (see Corollary 7.3). For this analysis, the dominating number
$\mathfrak {d}$
will be relevant, ultimately because of Hechler’s discovery that
$\mathsf {Exp}(\mathbb {Q})=\mathfrak {d}$
(see Corollary 3.3). In particular, when
$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$
, it will follow that
$\mathsf {Exp}(X)=\kappa $
for every countable dense subspace of
$2^\kappa $
(see Corollary 7.5). Finally, in Section 8, we will show that a scattered space of weight
$\kappa $
has pseudocharacter at most
$\kappa $
in any compactification (see Theorem 8.2). While this result seems to be of independent interest, it will allow us to calculate the realcompactness number of an arbitrary (that is, not necessarily crowded) countable space (see Corollary 8.4).
2. Preliminaries and terminology
Our reference for topological notions is [Reference Engelking3], except regarding cardinal functions, for which we refer to [Reference Juhász6]. Although our definition of realcompactness differs from the one given by Engelking, these two definitions are equivalent by [Reference Engelking3, Theorem 3.11.3]. For a more general treatment of
$\mathsf {Exp}$
and some history, we refer to the articles [Reference van Douwen2, Section 8] and [Reference Mrówka10], from which this notation was taken. However, the terminology “realcompactness number” is new.
Given a space X, make the following definitions:
-
•
$\mathsf {kc}(X)=\mathsf {min}\{|\mathcal {K}|:\mathcal {K}\text { is a cover of }X\text { consisting of compact sets}\}$
, -
•
$\mathsf {kc}^\ast (X)=\mathsf {kc}(\beta X\setminus X)$
.
It is clear that
$\mathsf {kc}^\ast $
is related to several well-studied notions as follows:
-
•
$\mathsf {kc}^\ast (X)=0$
iff X is compact, -
•
$\mathsf {kc}^\ast (X)=1$
iff X is locally compact but not compact, -
•
$\mathsf {kc}^\ast (X)=\omega $
iff X is Čech-complete but not locally compact.
The following simple but very convenient result shows that the value of
$\mathsf {kc}^\ast $
does not depend on the choice of compactification.
Proposition 2.1. Let X be a space, and let
$\gamma X$
be a compactification of X. Then
$\mathsf {kc}^\ast (X)=\mathsf {kc}(\gamma X\setminus X)$
.
Proof Using [Reference Engelking3, Theorem 3.5.7 and Corollary 3.6.6], it is possible to obtain a perfect surjection
$f:\beta X\setminus X\longrightarrow \gamma X\setminus X$
. Since f and
$f^{-1}$
both preserve compactness (see [Reference Engelking3, Theorem 3.7.2]), the desired result easily follows.
A space is crowded if it is non-empty and it has no isolated points. We will use
$\mathbb {R}$
to denote the space of real numbers with its standard topology, and
$\mathbb {Q}$
to denote the subspace of
$\mathbb {R}$
consisting of the rational numbers. We will write
$X\approx Y$
to mean that the spaces X and Y are homeomorphic. Given a space X, recall that the Cantor–Bendixson derivative of X is defined as follows for every ordinal
$\xi $
:
-
•
$X^{(0)}=X$
, -
•
$X^{(\xi +1)}=X^{(\xi )}\setminus \{x\in X^{(\xi )}:x\text { is isolated in }X^{(\xi )}\}$
, -
•
$X^{(\xi )}=\bigcap _{\xi '<\xi }X^{(\xi ')}$
, if
$\xi $
is a limit ordinal.
Define the perfect kernel of X as
$\mathsf {ker}(X)=X^{(\xi )}$
, where
$\xi $
is the smallest ordinal such that
$X^{(\xi )}=X^{(\xi +1)}$
. A space is scattered if
$\mathsf {ker}(X)=\varnothing $
.
The weight of a space X, denoted by
$\mathsf {w}(X)$
, is the maximum between
$\omega $
and the smallest cardinal
$\kappa $
such that there exists a base for X of size
$\kappa $
. Given a space X and
$x\in X$
, define the character of X at x, denoted by
$\chi (x,X)$
, to be the maximum between
$\omega $
and the smallest cardinal
$\kappa $
such that there exists a local base for X at x of size
$\kappa $
. Given a space X and
$S\subseteq X$
, define the pseudocharacter of X at S, denoted by
$\psi (S,X)$
, to be the maximum between
$\omega $
and the smallest cardinal
$\kappa $
such that there exists a collection
$\mathcal {O}$
of size
$\kappa $
consisting of open subsets of X such that
$\bigcap \mathcal {O}=S$
. When
$S=\{x\}$
, we will simply speak of the pseudocharacter of X at x, and denote it by
$\psi (x,X)$
.
Our reference for set-theoretic notions is [Reference Kunen8]. For convenience, we will assume that all partial ordersFootnote
2
have a maximum, which we denote by
$\mathbb {1}$
. Assume that a partial order
$\mathbb {P}$
is given. A subset
$\mathcal {C}$
of
$\mathbb {P}$
is centered if for every
$F\in [\mathcal {C}]^{<\omega }$
there exists
$q\in \mathbb {P}$
such that
$q\leq p$
for every
$p\in F$
. Recall that
$\mathbb {P}$
is
$\sigma $
-centered if there exist centered subsets
$\mathcal {C}_n$
of
$\mathbb {P}$
for
$n\in \omega $
such that
$\mathbb {P}=\bigcup _{n\in \omega }\mathcal {C}_n$
. We will say that
$\mathcal {F}\subseteq \mathbb {P}$
is a filter on
$\mathbb {P}$
if the following conditions hold:
-
•
$\mathbb {1}\in \mathcal {F}$
. -
• If
$p\in \mathcal {F}$
and
$p\leq q\in \mathbb {P}$
then
$q\in \mathcal {F}$
. -
• If
$p,q\in \mathcal {F}$
then there exists
$r\in \mathcal {F}$
such that
$r\leq p$
and
$r\leq q$
.
An ultrafilter is a filter that is maximal with respect to inclusion. It is easy to see that, when
$\mathbb {P}=\mathbb {B}\setminus \{\mathbb {0}\}$
for some boolean algebra
$\mathbb {B}$
, a filter
$\mathcal {F}$
on
$\mathbb {P}$
is an ultrafilter iff for every
$a\in \mathbb {P}$
either
$a\in \mathcal {F}$
of
$a^c\in \mathcal {F}$
.
Assume that a countably infinite set
$\Omega $
is given. A subset
$\mathcal {C}$
of
$[\Omega ]^\omega $
has the strong finite intersection property (briefly, the
$\mathsf {SFIP}$
) if
$C_0\cap \cdots \cap C_n$
is infinite whenever
$n\in \omega $
and
$C_0,\ldots ,C_n\in \mathcal {C}$
. We will write
$A\subseteq ^\ast B$
to mean that
$A\setminus B$
is finite. We will say that
$A\subseteq \Omega $
is a pseudointersection of
$\mathcal {C}\subseteq [\Omega ]^\omega $
if A is infinite and
$A\subseteq ^\ast C$
for all
$C\in\mathcal{C}$
. We will denote by
$\mathfrak {p}$
the pseudointersection number, that is the minimum cardinality of a subset of
$[\omega ]^\omega $
with the
$\mathsf {SFIP}$
and no pseudointersection. We will denote by
$\mathfrak {d}$
the dominating number, that is the minimum cardinality of a subset
$\mathcal {D}$
of
$\omega ^\omega $
such that for every
$f\in \omega ^\omega $
there exists
$g\in \mathcal {D}$
such that
$f(n)\leq g(n)$
for all but finitely many
$n\in \omega $
. We will say that
$\mathcal {A}\subseteq [\Omega ]^\omega $
is an independent family on
$\Omega $
if
$A_0\cap \cdots \cap A_m\cap (\omega \setminus B_0)\cap \cdots \cap (\omega \setminus B_n)$
is infinite whenever
$m,n\in \omega $
and
$A_0,\ldots ,A_m,B_0,\ldots ,B_n\in \mathcal {A}$
are distinct. If the set
$\Omega $
is not mentioned, we will assume that
$\Omega =\omega $
. The following result is credited to Nyikos by Matveev in [Reference Matveev9].
Proposition 2.2 (Nyikos)
There exists an independent family of size
$\mathfrak {p}$
with no pseudointersection.
Proof Clearly, it will be enough to construct such an independent family on
$\omega \times \omega $
. Fix an independent family
$\mathcal {A}$
of size
$\mathfrak {p}$
(see [Reference Kunen8, Exercise III.1.35]), and a subset
$\mathcal {C}$
of
$[\omega ]^\omega $
of size
$\mathfrak {p}$
with the
$\mathsf {SFIP}$
and no pseudointersection. Let
$\mathcal {A}=\{A_\xi :\xi <\mathfrak {p}\}$
and
$\mathcal {C}{\kern-1pt}={\kern-1pt}\{C_\xi :\xi {\kern-1pt}<{\kern-1pt}\mathfrak {p}\}$
be injective enumerations. Set
$\Delta ^+{\kern-1pt}={\kern-1pt}\{(m,n)\in \omega {\kern-1pt}\times{\kern-1pt} \omega :m{\kern-1pt}\leq{\kern-1pt} n\}$
. It is straightforward to verify that
$\{(A_\xi \times C_\xi )\cap \Delta ^+:\xi <\mathfrak {p}\}$
is an independent family on
$\omega \times \omega $
of size
$\mathfrak {p}$
with no pseudointersection.
3. Calculating the realcompactness number
The aim of this section is to develop tools to calculate the realcompactness number. We will limit ourselves to the context of Lindelöf spaces, as the theory is particularly pleasant in this case. Furthermore, since the focus of this article is on countable spaces, this level of generality will clearly suffice. We also remark that Theorem 3.2 generalizes [Reference van Douwen2, Lemma 8.19] from separable metrizable to Lindelöf. Corollary 3.3 gives the simplest non-trivial application of these methods, and it will be a crucial ingredient for the results of Section 7. This theorem first appeared in [Reference Hechler5], where it is proved using a result from [Reference Hechler4].
Lemma 3.1. Let Z be a space, let X be a Lindelöf subspace of Z, and let
$K\subseteq Z\setminus X$
be compact. Then there exists a continuous function
$f:Z\longrightarrow [0,1]$
such that
$f(z)=0$
for every
$z\in K$
and
$f(z)>0$
for every
$z\in X$
.
Proof For every
$x\in X$
, fix a continuous function
$f_x:Z\longrightarrow [0,1]$
such that
$f_x(z)=0$
for every
$z\in K$
and
$f_x(x)=1$
. By continuity, for every
$x\in X$
there exists an open neighborhood
$U_x$
of x in Z such that
$f_x(z)>0$
for every
$z\in U_x$
. Since X is Lindelöf, there exist
$x_n\in X$
for
$n\in \omega $
that
$X\subseteq \bigcup _{n\in \omega }U_{x_n}$
. Finally, define
$f:Z\longrightarrow [0,1]$
by setting
$$ \begin{align*}f(z)=\sum_{n\in\omega}\frac{f_{x_n}(z)}{2^{n+1}} \end{align*} $$
for
$z\in Z$
. It is straightforward to check that f is as desired.
Theorem 3.2. Let X be a Lindelöf space. Then
$$ \begin{align*}\mathsf{Exp}(X)=\mathsf{max}\{\mathsf{w}(X),\mathsf{kc}^\ast(X)\}. \end{align*} $$
Proof Set
$\kappa =\mathsf {Exp}(X)$
and
$\kappa '=\mathsf {max}\{\mathsf {w}(X),\mathsf {kc}^\ast (X)\}$
. We begin by showing that
$\kappa \leq \kappa '$
. By [Reference Engelking3, Theorem 3.5.2], we can fix a compactification
$\gamma X$
of X such that
$\mathsf {w}(\gamma X)=\mathsf {w}(X)$
. By Proposition 2.1, there exist compact subsets
$K_\xi $
of
$\gamma X\setminus X$
for
$\xi \in \kappa '$
such that
$\bigcup _{\xi \in \kappa '}K_\xi =\gamma X\setminus X$
. By Lemma 3.1, there exist continuous functions
$f_\xi :\gamma X\longrightarrow [0,1]$
for
$\xi \in \kappa '$
such that
$f_\xi (z)=0$
for every
$z\in K_\xi $
and
$f_\xi (z)>0$
for every
$z\in X$
. Set
$\mathcal {F}=\{f_\xi :\xi \in \kappa '\}$
. Also fix a collection
$\mathcal {G}$
of size at most
$\mathsf {w}(\gamma X)=\mathsf {w}(X)$
consisting of continuous functions
$g:\gamma X\longrightarrow \mathbb {R}$
that separates points of
$\gamma X$
. In other words, whenever
$x,y\in \gamma X$
are distinct, there exists
$g\in \mathcal {G}$
such that
$g(x)\neq g(y)$
. Define
$\phi :\gamma X\longrightarrow \mathbb {R}^{\mathcal {F}\cup \mathcal {G}}$
by setting
$\phi (z)(f)=f(z)$
for
$z\in \gamma X$
and
$f\in \mathcal {F}\cup \mathcal {G}$
. For notational convenience, we will identify
$\mathbb {R}^{\mathcal {F}\cup \mathcal {G}}$
with
$\mathbb {R}^{\mathcal {F}}\times \mathbb {R}^{\mathcal {G}}$
in the obvious way.
Notice that
$\phi $
is continuous because each one of its coordinates is continuous. Furthermore, our choice of
$\mathcal {G}$
will guarantee that
$\phi $
is injective, hence
$\phi $
is an embedding by compactness. Therefore
$$ \begin{align*}X\approx\phi[X]=\phi[\gamma X]\cap\left((0,\infty)^{\mathcal{F}}\times\mathbb{R}^{\mathcal{G}}\right). \end{align*} $$
Since
$(0,\infty )^{\mathcal {F}}\times \mathbb {R}^{\mathcal {G}}\approx \mathbb {R}^{\mathcal {F}\cup \mathcal {G}}$
and
$\phi [\gamma X]$
is compact, it follows that
$\phi [X]\approx X$
is homeomorphic to a closed subspace of
$\mathbb {R}^{\mathcal {F}\cup \mathcal {G}}$
. But
$|\mathcal {F}\cup \mathcal {G}|\leq \kappa '$
by construction, hence
$\kappa \leq \kappa '$
.
To finish the proof, we will show that
$\kappa '\leq \kappa $
. Assume without loss of generality that X is a closed subspace of
$(0,1)^\kappa $
, and let
$Z=\mathsf {cl}(X)$
, where the closure is taken in
$[0,1]^\kappa $
. Observe that Z is a compactification of X. Denote by
$\pi _\xi :[0,1]^\kappa \longrightarrow [0,1]$
for
$\xi \in \kappa $
the natural projection on the
$\xi $
th coordinate. Since X is closed in
$(0,1)^\kappa $
, for every
$z\in Z\setminus X$
there exists
$\xi \in \kappa $
such that
$z(\xi )\in \{0,1\}$
. In other words,
$$ \begin{align*}Z\setminus X=\bigcup_{\xi\in\kappa}(\pi_\xi^{-1}[\{0,1\}]\cap Z). \end{align*} $$
But each
$\pi _\xi ^{-1}[\{0,1\}]\cap Z$
is compact, hence
$\mathsf {kc}^\ast (X)\leq \kappa $
by Proposition 2.1. Since clearly
$\mathsf {w}(X)\leq \mathsf {w}\big ((0,1)^\kappa \big )=\kappa $
, it follows that
$\kappa '\leq \kappa $
, as desired.
Corollary 3.3 (Hechler)
$\mathsf {Exp}(\mathbb {Q})=\mathfrak {d}$
.
Proof By [Reference Engelking3, Exercise 6.2.A(e)], it is possible to fix a compactification
$\gamma \mathbb {Q}$
of
$\mathbb {Q}$
such that
$\gamma \mathbb {Q}\approx 2^\omega $
. It follows from [Reference Engelking3, Theorem 4.3.23 and Exercise 6.2.A(b)] that
$\gamma \mathbb {Q}\setminus \mathbb {Q}\approx \omega ^\omega $
. Since
$\mathsf {kc}(\omega ^\omega )=\mathfrak {d}$
(see [Reference van Douwen2, Theorem 8.2]), this shows that
$\mathsf {kc}^\ast (\mathbb {Q})=\mathfrak {d}$
by Proposition 2.1. An application of Theorem 3.2 concludes the proof.
We conclude this section with a result which will enable us to find a bound on the realcompactness number by finding a suitable finite decomposition of the space in question. It will be used in the proofs of Theorem 6.1 and Corollary 8.4.
Proposition 3.4. Let X be a Lindelöf space. Assume that
$n\in \omega $
and
$X_0,\ldots ,X_n$
are Lindelöf subspaces of X such that
$X=X_0\cup \cdots \cup X_n$
. Then
$$ \begin{align*}\mathsf{Exp}(X)\leq\mathsf{max}\{\mathsf{Exp}(X_0),\ldots,\mathsf{Exp}(X_n),\mathsf{w}(X)\}. \end{align*} $$
Proof Set
$\kappa =\mathsf {max}\{\mathsf {Exp}(X_0),\ldots ,\mathsf {Exp}(X_n),\mathsf {w}(X)\}$
. By [Reference Engelking3, Theorem 3.5.2], we can fix a compactification
$\gamma X$
of X such that
$\mathsf {w}(\gamma X)=\mathsf {w}(X)$
. Throughout this proof, we will use
$\mathsf {cl}$
to denote closure in
$\gamma X$
.
Since each
$X_i$
is Lindelöf, by Theorem 3.2 it is possible to find compact sets
$K_{i,\xi }$
for
$0\leq i\leq n$
and
$\xi \in \kappa $
such that
$$ \begin{align*}\mathsf{cl}(X_i)\setminus X_i=\bigcup_{\xi\in\kappa}K_{i,\xi} \end{align*} $$
for each i. On the other hand, using the compactness of each
$\mathsf {cl}(X_i)$
and the fact that
$\mathsf {w}(\gamma X)\leq \kappa $
, it is easy to see that each
$\psi (\mathsf {cl}(X_i),\gamma X)\leq \kappa $
. This means that there exist open subsets
$U_{i,\xi }$
of
$\gamma X_i$
for
$0\leq i\leq n$
and
$\xi \in \kappa $
such that
$$ \begin{align*}\gamma X\setminus\mathsf{cl}(X_i)=\bigcup_{\xi\in\kappa}(\gamma X\setminus U_{i,\xi}) \end{align*} $$
for each i. In conclusion, set
$$ \begin{align*}\mathcal{K}_i=\{K_{i,\xi}:\xi\in\kappa\}\cup\{\gamma X\setminus U_{i,\xi}:\xi\in\kappa\} \end{align*} $$
for
$0\leq i\leq n$
, and observe that each
$\mathcal {K}_i$
is a cover of
$\gamma X\setminus X_i$
of size at most
$\kappa $
consisting of compact sets. Therefore, the sets of the form
$K_0\cap \cdots \cap K_n$
, where each
$K_i\in \mathcal {K}_i$
, will cover
$\gamma X\setminus X$
. Since there are at most
$\kappa $
sets of this form, and they are clearly compact, a further application of Theorem 3.2 will conclude the proof.
4. The lower bound
In this section, we will give a lower bound on
$\mathsf {Exp}(X)$
for a countable crowded space X (see Corollary 4.3). Chronologically, this is the first result that we obtained, and it motivated us to look for the examples constructed in the later sections.
Theorem 4.1. Let
$\kappa <\mathfrak {p}$
be an infinite cardinal, and let X be a countable crowded subspace of
$\omega ^\kappa $
. Then X is not closed in
$\omega ^\kappa $
.
Proof Define
$$ \begin{align*}\mathbb{P}=\{x\upharpoonright a:x\in X\text{ and }a\in [\kappa]^{<\omega}\}. \end{align*} $$
Given
$s,t\in \mathbb {P}$
, declare
$s\leq t$
if
$s\supseteq t$
, and observe that
$\mathbb {P}$
is a partial order. Also notice that
$\mathbb {P}$
is
$\sigma $
-centered because
$$ \begin{align*}\mathbb{P}=\bigcup_{x\in X}\{x\upharpoonright a:a\in [\kappa]^{<\omega}\}. \end{align*} $$
Given
$x\in X$
and
$a\in [\kappa ]^{<\omega }$
, define:
-
•
$D_x=\{s\in \mathbb {P}:s(\xi )\neq x(\xi )\text { for some }\xi \in \mathsf {dom}(s)\}$
. -
•
$D_a=\{s\in \mathbb {P}:s=x\upharpoonright b\text { for some }x\in X\text { and }b\in [\kappa ]^{<\omega }\text { such that }b\supseteq a\}$
.
Using the fact that X is crowded, one sees that each
$D_x$
is dense in
$\mathbb {P}$
. On the other hand, it is trivial to check that each
$D_a$
is dense in
$\mathbb {P}$
. Since
$\kappa <\mathfrak {p}$
, Bell’s theorem (see [Reference Kunen8, Theorem III.3.61])Footnote
3
guarantees the existence of a filter G on
$\mathbb {P}$
that meets all of these dense sets. It is easy to realize that
$\bigcup G\in \mathsf {cl}(X)\setminus X$
, where
$\mathsf {cl}$
denotes closure in
$\omega ^\kappa $
.
Corollary 4.2. Let
$\kappa <\mathfrak {p}$
be an infinite cardinal, and let
$Z_\xi $
for
$\xi \in \kappa $
be Polish spaces. Set
$Z=\prod _{\xi \in \kappa }Z_\xi $
. If X is a countable crowded subspace of Z then X is not closed in Z.
Proof Let X be a countable crowded subspace of Z. Assume, in order to get a contradiction, that X is closed in Z. Notice that the existence of X implies that
$|Z_\xi |\geq 2$
for infinitely many values of
$\xi $
. Therefore, by taking suitable countably infinite products, we can assume without loss of generality, that each
$Z_\xi $
is crowded. Given
$\xi \in \kappa $
, denote by
$\pi _\xi :Z\longrightarrow Z_\xi $
the natural projection on the
$\xi $
th coordinate. By [Reference Engelking3, Exercise 7.4.17], it is possible to obtain a zero-dimensional dense
$\mathsf {G}_\delta $
subspace
$Z^{\prime }_\xi $
of
$Z_\xi $
for each
$\xi $
such that each
$\pi _\xi [X]\subseteq Z^{\prime }_\xi $
. By removing a countable dense subset of
$Z^{\prime }_\xi $
disjoint from
$\pi _\xi [X]$
, we can assume that each
$Z^{\prime }_\xi $
has no non-empty compact open subsets. It follows from [Reference Engelking3, Theorem 4.3.23 and Exercise 6.2.A(b)] that
$Z^{\prime }_\xi \approx \omega ^\omega $
for each
$\xi $
. Finally, the fact that X is a closed subset of
$$ \begin{align*}\prod_{\xi\in\kappa}Z^{\prime}_\xi\approx (\omega^\omega)^\kappa\approx\omega^\kappa \end{align*} $$
contradicts Theorem 4.1.
Corollary 4.3. If X is a countable crowded space then
$\mathsf {Exp}(X)\geq \mathfrak {p}$
.
5. A countable crowded space with realcompactness number
$\mathfrak {p}$
In this section, we will construct the space promised by Theorem 1.1 in the case
$\kappa =\mathfrak {p}$
. Once this is established, the general case will follow without much trouble (see Section 6). We will assume some familiarity with the basic theory of boolean algebras and Stone duality (see [Reference Koppelberg, Donald Monk and Bonnet7], or [Reference Kunen8, Section III.4] for a concise exposition). While the core of the construction is given in the proof of Theorem 5.4, we will need several technical preliminaries concerning embeddings and filters.
Let partial orders
$\mathbb {P}$
and
$\mathbb {P}'$
be given. Inspired by [Reference Kunen8, Definition III.3.65], we will say that
$i:\mathbb {P}\longrightarrow \mathbb {P}'$
is a pleasant embedding
Footnote
4
if the following conditions are satisfied:
-
(1)
$i(\mathbb {1})=\mathbb {1}$
, -
(2)
$\forall p,q\in \mathbb {P}\,\big (p\leq q\rightarrow i(p)\leq i(q)\big )$
, -
(3)
$\forall p,q\in \mathbb {P}\,\big (p\perp q\leftrightarrow i(p)\perp i(q)\big )$
.
We will say that i is a dense embedding if it satisfies all of the above conditions plus the following:
-
(4)
$i[\mathbb {P}]$
is dense in
$\mathbb {P}'$
.
Also recall that
$\mathbb {P}$
is separative if for all
$p,q\in \mathbb {P}$
such that
$p\nleq q$
there exists
$r\in \mathbb {P}$
such that
$r\leq p$
and
$r\perp q$
.
We will say that
$\mathbb {P}$
is meet-friendly if whenever
$p,q\in \mathbb {P}$
are compatible the set
$\{p,q\}$
has a greatest lower bound, which we will denote by
$p\wedge q$
. In this case, it is clear that
$p\wedge q\in \mathcal {F}$
whenever
$\mathcal {F}$
is a filter on
$\mathbb {P}$
and
$p,q\in \mathcal {F}$
. Notice that
$\mathbb {P}$
is meet-friendly iff every centered finite subset
$\{p_0,\ldots ,p_n\}$
of
$\mathbb {P}$
has a greatest lower bound, which we will denote by
$p_0\wedge \cdots \wedge p_n$
. An important example of meet-friendly partial-order is given by
$\mathbb {B}\setminus \{\mathbb {0}\}$
whenever
$\mathbb {B}$
is a boolean algebra. When
$\mathbb {P}$
is meet-friendly and
$\mathcal {C}$
is a non-empty centered subset of
$\mathbb {P}$
, it makes sense to consider the filter
$\mathcal {F}$
generated by
$\mathcal {C}$
(that is, the smallest filter on
$\mathbb {P}$
containing
$\mathcal {C}$
). In fact, it is easy to check that this filter has the following familiar form:
$$ \begin{align*}\mathcal{F}=\{p\in\mathbb{P}:p_0\wedge\cdots\wedge p_n\leq p\text{ for some }n\in\omega\text{ and }p_0,\ldots,p_n\in\mathcal{C}\}. \end{align*} $$
Finally, when
$\mathbb {P}$
and
$\mathbb {P}'$
are meet-friendly, we will say that a pleasant embedding
$i:\mathbb {P}\longrightarrow \mathbb {P}'$
is meet-preserving if the following condition is satisfied:
-
(5)
$\forall p,q\in \mathbb {P}\,\big (p\not \perp q\rightarrow i(p\wedge q)=i(p)\wedge i(q)\big )$
.
Lemma 5.1. Let
$\mathbb {P}$
be a meet-friendly partial order, and let
$\mathcal {F}$
be a filter on
$\mathbb {P}$
. Then the following conditions are equivalent
$:$
-
(A)
$\mathcal {F}$
is an ultrafilter, -
(B)
$\forall p\in \mathbb {P}\setminus \mathcal {F}\,\exists q\in \mathcal {F}\, (p\perp q)$
.
Proof In order to prove that
$(\text {A})\rightarrow (\text {B})$
, assume that
$p\in \mathbb {P}\setminus \mathcal {F}$
is such that
$p\not \perp q$
for every
$q\in \mathcal {F}$
. Since
$\mathbb {P}$
is meet-friendly, it makes sense to consider
$$ \begin{align*}\mathcal{G}=\mathcal{F}\cup\{r\in\mathbb{P}:r\geq p\wedge q\text{ for some }q\in\mathcal{F}\}. \end{align*} $$
It is straightforward to check that
$\mathcal {G}\supsetneq \mathcal {F}$
is a filter on
$\mathbb {P}$
, hence
$\mathcal {F}$
is not an ultrafilter. In order to prove that
$(\text {B})\rightarrow (\text {A})$
, assume that
$\mathcal {G}\supsetneq \mathcal {F}$
is a filter on
$\mathbb {P}$
. It is clear that any
$p\in \mathcal {G}\setminus \mathcal {F}$
will witness the failure of condition
$(\text {B})$
.
Lemma 5.2. Let
$\mathbb {P}$
and
$\mathbb {P}'$
be meet-friendly partial orders, and let
$i:\mathbb {P}\longrightarrow \mathbb {P}'$
be a meet-preserving pleasant embedding. If
$\mathcal {G}$
is a filter on
$\mathbb {P}'$
then
$i^{-1}[\mathcal {G}]$
is a filter on
$\mathbb {P}$
.
Proof Let
$\mathcal {G}$
be a filter on
$\mathbb {P}'$
, and set
$\mathcal {F}=i^{-1}[\mathcal {G}]$
. The fact that
$\mathbb {1}\in \mathcal {F}$
follows from condition
$(1)$
. In order to show that
$\mathcal {F}$
is upward-closed, let
$p\in \mathcal {F}$
and
$q\in \mathbb {P}$
be such that
$p\leq q$
. Since
$i(p)\in \mathcal {G}$
and
$i(p)\leq i(q)$
by condition
$(2)$
, we must have
$i(q)\in \mathcal {G}$
. Hence
$q\in \mathcal {F}$
, as desired. To complete the proof, pick
$p,q\in \mathcal {F}$
. Since
$i(p),i(q)\in \mathcal {G}$
, we must have
$i(p)\not \perp i(q)$
, hence
$p\not \perp q$
by condition
$(3)$
. Therefore, we can consider
$p\wedge q$
, and observe that
$i(p\wedge q)=i(p)\wedge i(q)\in \mathcal {G}$
by condition
$(5)$
. It follows that
$p\wedge q\in \mathcal {F}$
, as desired.
Lemma 5.3. Let
$\mathbb {P}$
be a meet-friendly partial order, let
$\mathbb {B}$
be a boolean algebra, and let
$i:\mathbb {P}\longrightarrow \mathbb {B}\setminus \{\mathbb {0}\}$
be a pleasant embedding. Assume that
$i[\mathbb {P}]$
generates
$\mathbb {B}$
as a boolean algebra. If
$\mathcal {U}$
is an ultrafilter on
$\mathbb {P}$
then
$i[\mathcal {U}]$
generates an ultrafilter on
$\mathbb {B}\setminus \{\mathbb {0}\}$
.
Proof Pick an ultrafilter
$\mathcal {U}$
on
$\mathbb {P}$
. Since
$i[\mathcal {U}]$
is centered by condition
$(2)$
, we can consider the filter
$\mathcal {V}$
on
$\mathbb {B}$
generated by
$i[\mathcal {U}]$
. Define
$$ \begin{align*}\mathbb{B}'=\{a\in\mathbb{B}:i(p)\leq a\text{ or }i(p)\leq a^c\text{ for some }p\in\mathcal{U}\}. \end{align*} $$
Claim 1.
$\mathbb {B}'$
is a boolean subalgebra of
$\mathbb {B}$
.
Proof The fact that
$\mathbb {1}\in \mathbb {B}'$
follows from condition
$(1)$
. Furthermore, it is clear that
$a\in \mathbb {B}'$
iff
$a^c\in \mathbb {B}'$
. In order to show that
$\mathbb {B}'$
is closed under
$\vee $
, pick
$a,b\in \mathbb {B}'$
. If
$i(p)\leq a$
or
$i(p)\leq b$
for some
$p\in \mathcal {U}$
, then it is clear that
$a\vee b\in \mathbb {B}'$
. So let
$p,q\in \mathcal {U}$
be such that
$i(p)\leq a^c$
and
$i(q)\leq b^c$
. Since
$\mathcal {U}$
is a filter, we can pick
$r\in \mathcal {U}$
such that
$r\leq p$
and
$r\leq q$
. It follows from condition
$(2)$
that
$$ \begin{align*}i(r)\leq i(p)\wedge i(q)\leq a^c\wedge b^c=(a\vee b)^c, \end{align*} $$
which shows that
$a\vee b\in \mathbb {B}'$
, as desired.
Claim 2.
$i[\mathbb {P}]\subseteq \mathbb {B}'$
.
Proof Pick
$a\in i[\mathbb {P}]$
, and let
$p\in \mathbb {P}$
be such that
$i(p)=a$
. If
$p\in \mathcal {U}$
then clearly
$a\in \mathbb {B}'$
, so assume that
$p\notin \mathcal {U}$
. Since
$\mathbb {P}$
is meet-friendly and
$\mathcal {U}$
is an ultrafilter, Lemma 5.1 yields
$q\in \mathcal {U}$
such that
$p\perp q$
. Notice that
$i(p)\perp i(q)$
by condition
$(3)$
, hence
$i(q)\leq i(p)^c=a^c$
. This shows that
$a\in \mathbb {B}'$
, as desired.
Since
$i[\mathbb {P}]$
generates
$\mathbb {B}$
as a boolean algebra, it follows from Claims 1 and 2 that
$\mathbb {B}'=\mathbb {B}$
. By the definition of
$\mathbb {B}'$
, this shows that
$\mathcal {V}$
is an ultrafilter on
$\mathbb {B}\setminus \{\mathbb {0}\}$
.
Given
$a,b\in [\omega ]^{<\omega }$
, we will write
$a\preccurlyeq b$
to mean
$a\subseteq b$
and
$b\setminus a\subseteq \omega \setminus \mathsf {max}(a)$
. We will also write
$a\prec b$
to mean
$a\preccurlyeq b$
and
$a\neq b$
. Given a subset
$\mathcal {C}$
of
$[\omega ]^\omega $
with the
$\mathsf {SFIP}$
, define
$$ \begin{align*}\mathbb{P}(\mathcal{C})=\{(a,F):a\in [\omega]^{<\omega}\text{ and }F\in [\mathcal{C}]^{<\omega}\}. \end{align*} $$
Order
$\mathbb {P}(\mathcal {C})$
by declaring
$(a,F)\leq (b,G)$
if the following conditions hold:
-
•
$b\preccurlyeq a$
, -
•
$G\subseteq F$
, -
•
$a\setminus b\subseteq \bigcap G$
,
where we make the convention that
$\bigcap \varnothing =\omega $
. This is of course the standard partial order that generically produces a pseudointersection of
$\mathcal {C}$
. We remark that
$\mathbb {P}(\mathcal {C})$
is always meet-friendly. In fact, given
$(a,F),(b,G)\in \mathbb {P}(\mathcal {C})$
such that
$(a,F)\not \perp (b,G)$
, it is easy to realize that
$(a\cup b,F\cup G)$
is the greatest lower bound of
$\{(a,F),(b,G)\}$
.
Theorem 5.4. There exists a countable crowded space X such that
$\mathsf {Exp}(X)=\mathfrak {p}$
.
Proof By Proposition 2.2, we can fix an independent family
$\mathcal {A}$
of size
$\mathfrak {p}$
with no pseudointersection. Without loss of generality, assume that for every
$n\in \omega $
there exists
$A\in \mathcal {A}$
such that
$n\notin A$
. Set
$\mathbb {P}=\mathbb {P}(\mathcal {A})$
. Given
$a\in [\omega ]^{<\omega }$
, denote by
$\mathcal {U}_a$
the filter on
$\mathbb {P}$
generated by
$\{(a,F):F\in [\mathcal {A}]^{<\omega }\}$
.
Claim 1. Each
$\mathcal {U}_a$
is an ultrafilter on
$\mathbb {P}$
.
Proof Fix
$a\in [\omega ]^{<\omega }$
. Pick
$(b,G)\in \mathbb {P}$
that is compatible with
$(a,F)$
for every
$F\in [\mathcal {A}]^{<\omega }$
. We will show that
$(b,G)\in \mathcal {U}_a$
, which will be sufficient by Lemma 5.1. It is easy to realize that either
$a\prec b$
or
$b\preccurlyeq a$
. First assume that
$a\prec b$
, and pick
$n\in b\setminus a$
. By our choice of
$\mathcal {A}$
, there exists
$A\in \mathcal {A}$
such that
$n\notin A$
. It is clear that
$(a,\{A\})\perp (b,G)$
, which shows that this case is impossible. Therefore
$b\preccurlyeq a$
. Pick
$(c,H)\in \mathbb {P}$
such that
$(c,H)\leq (b,G)$
and
$(c,H)\leq (a,G)$
. Observe that
$a\setminus b\subseteq c\setminus b\subseteq \bigcap G$
, hence
$(a,G)\leq (b,G)$
. It follows that
$(b,G)\in \mathcal {U}_a$
, as desired.
Claim 2.
$\mathbb {P}$
is separative.
Proof Pick
$(a,F),(b,G)\in \mathbb {P}$
such that
$(a,F)\nleq (b,G)$
. We will find
$(c,H)\leq (a,F)$
that is incompatible with
$(b,G)$
. First assume that
$b\nsubseteq a$
. Pick
$n\in \bigcap F$
big enough so that
$n>\mathsf {max}(a)$
and
$n>\mathsf {max}(b)$
. It is easy to realize that
$(a\cup \{n\},F)\leq (a,F)$
and that
$(a\cup \{n\},F)\perp (b,G)$
.
Next, assume that
$b\subseteq a$
but
$b\not \preccurlyeq a$
. This means that there exists
$n\in a\setminus b$
such that
$n<\mathsf {max}(b)$
, hence
$(a,F)\perp (b,G)$
. Next, assume that
$b\preccurlyeq a$
but there exists
$n\in a\setminus b$
such that
$n\notin \bigcap G$
. It is clear that
$(a,F)\perp (b,G)$
in this case as well.
Finally, assume that
$b\preccurlyeq a$
,
$a\setminus b\subseteq \bigcap G$
, but
$G\nsubseteq F$
. Let
$A\in G\setminus F$
. Since
$\mathcal {A}$
is an independent family, it is possible to pick
$n\in (\omega \setminus A)\cap \bigcap F$
big enough so that
$n>\mathsf {max}(a)$
. It is clear that
$(a\cup \{n\},F)\leq (a,F)$
and that
$(a\cup \{n\},F)\perp (b,G)$
.
Given
$p\in \mathbb {P}$
, we will use the notation
$p\!\downarrow \,=\{q\in \mathbb {P} :q\leq p\}$
. Declare
$U\subseteq \mathbb {P}$
to be open if
$p\!\downarrow \,\subseteq U$
for every
$p\in U$
. We will denote by
$\mathsf {RO}(\mathbb {P})$
the regular open algebra of
$\mathbb {P}$
according to this topology (see [Reference Kunen8, Definition III.4.6]). Define
$i:\mathbb {P}\longrightarrow \mathsf {RO}(\mathbb {P})\setminus \{\mathbb {0}\}$
by setting
$i(p)=p\!\downarrow \,$
for
$p\in \mathbb {P}$
. It follows from Claim 2 and [Reference Kunen8, Lemma III.4.8 and Exercise III.4.15] that i is a well-defined dense embedding, and it is straightforward to verify that i is meet-preserving. Furthermore, by [Reference Kunen8, Exercise III.4.11], the following stronger form of condition
$(2)$
holds:
-
(2′)
$\forall p,q\in \mathbb {P}\,\big (p\leq q\leftrightarrow i(p)\leq i(q)\big )$
.
Let
$\mathbb {B}$
be the boolean subalgebra of
$\mathsf {RO}(\mathbb {P})$
generated by
$i[\mathbb {P}]$
, and let
$$ \begin{align*}Z=\{\mathcal{V}:\mathcal{V}\text{ is an ultrafilter on }\mathbb{B}\setminus\{\mathbb{0}\}\} \end{align*} $$
denote the Stone space of
$\mathbb {B}$
. Given
$b\in \mathbb {B}$
, we will denote by
$[b]=\{\mathcal {V}\in Z:b\in \mathcal {V}\}$
the corresponding basic clopen subset of Z. By Claim 1 and Lemma 5.3, each
$i[\mathcal {U}_a]$
generates an ultrafilter on
$\mathbb {B}\setminus \{\mathbb {0}\}$
, which we will denote by
$\mathcal {V}_a$
. Finally, set
$$ \begin{align*}X=\{\mathcal{V}_a:a\in [\omega]^{<\omega}\}. \end{align*} $$
Claim 3. Z is crowded.
Proof This is equivalent to showing that
$\mathbb {B}$
has no atoms. So pick
$b\in \mathbb {B}\setminus \{\mathbb {0}\}$
. By condition
$(4)$
, there exists
$p\in \mathbb {P}$
such that
$i(p)\leq b$
. Now choose any
$q\in \mathbb {P}$
such that
$q<p$
, and observe that
$\mathbb {0} <i(q)<i(p)\leq b$
by condition
$(2')$
. This shows that b is not an atom.
Claim 4. X is a countable dense subset of Z.
Proof The fact that X is countable is clear. In order to see that X is dense, pick a non-empty open subset U of Z. Without loss of generality, assume that
$U=[b]$
for some
$b\in \mathbb {B}\setminus \{\mathbb {0}\}$
. By condition
$(4)$
, we can actually assume that
$b=(a,F)\!\downarrow \,$
for some
$(a,F)\in \mathbb {P}$
. It is clear that
$b\in i[\mathcal {U}_a]\subseteq \mathcal {V}_a$
, hence
$\mathcal {V}_a\in [b]=U$
.
It follows from Claims 3 and 4 that X is a countable crowded space, and that Z is a compactification of X. Furthermore, it is easy to see that
$\mathsf {w}(X)\leq \mathsf {w}(Z)=|\mathbb {B}|=\mathfrak {p}$
, where
$|\mathbb {B}|\geq \mathfrak {p}$
holds by condition
$(2')$
. Since
$\mathsf {Exp}(X)\geq \mathfrak {p}$
by Corollary 4.3, the following claim will conclude the proof by Theorem 3.2.
Fix an enumeration
$\mathcal {A}=\{A_\xi :\xi \in \mathfrak {p}\}$
. Given
$\xi \in \mathfrak {p}$
, set
$$ \begin{align*}U_\xi=\bigcup_{a\in [\omega]^{<\omega}}[(a,\{A_\xi\})\!\downarrow\,], \end{align*} $$
and observe that each
$U_\xi $
is an open subset of Z.
Claim 5.
$X=\bigcap _{\xi \in \mathfrak {p}}U_\xi $
.
Proof The inclusion
$\subseteq $
is straightforward. In order to prove the inclusion
$\supseteq $
, pick
$\mathcal {V}\in \bigcap _{\xi \in \mathfrak {p}}U_\xi $
. This means that for every
$\xi \in \mathfrak {p}$
we can fix
$a_\xi \in [\omega ]^{<\omega }$
such that
$(a_\xi ,\{A_\xi\} )\!\downarrow \,\in \mathcal {V}$
. Set
$\mathcal {U}=i^{-1}[\mathcal {V}]$
, and observe that
$\mathcal {U}$
is a filter on
$\mathbb {P}$
by Lemma 5.2. Furthermore, it is clear that each
$(a_\xi ,\{A_\xi\} )\in \mathcal {U}$
.
Set
$a=\bigcup _{\xi \in \mathfrak {p}}a_\xi $
. First assume, in order to get a contradiction, that a is infinite. Using the fact that each
$(a_\xi ,\{A_\xi\} )\in \mathcal {U}$
, it is easy to verify that
$a\setminus \mathsf {max}(a_\xi )\subseteq A_\xi $
for each
$\xi $
. In other words, the set a is a pseudointersection of
$\mathcal {A}$
, which is a contradiction.
Therefore a is finite, hence we can fix
$\xi \in \mathfrak {p}$
such that
$a=a_\xi $
. We will show that
$\mathcal {U}=\mathcal {U}_a$
, which easily implies that
$\mathcal {V}=\mathcal {V}_a$
, thus concluding the proof. Since
$\mathcal {U}_a$
is an ultrafilter, it will be enough to show that
$\mathcal {U}_a\subseteq \mathcal {U}$
. So pick
$(a,F)\in \mathcal {U}_a$
, where
$F=\{A_{\xi _0},\ldots ,A_{\xi _k}\}$
. As in the proof that
$\mathbb {P}$
is meet-friendly, one sees that
$$ \begin{align*}(a,F\cup\{A_\xi\})=(a_{\xi_0},\{A_{\xi_0}\})\wedge\cdots\wedge (a_{\xi_k},\{A_{\xi_k}\})\wedge (a_{\xi},\{A_\xi\})\in\mathcal{U}, \end{align*} $$
which clearly implies
$(a,F)\in \mathcal {U}$
, as desired.
6. Bumping-up the weight
In this section, we will finally exhibit the examples promised in Theorem 1.1. The strategy is to start with the space given by Theorem 5.4, then artificially increase its weight. Since we will achieve this by adding a single point, Proposition 3.4 will guarantee that the realcompactness number will not grow more than we want it to.
Theorem 6.1. Let
$\kappa $
be a cardinal such that
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$
. Then there exists a countable crowded space
$X_\kappa $
such that
$\mathsf {Exp}(X_\kappa )=\kappa $
.
Proof By Theorem 5.4, we can fix a countable crowded space X with
$\mathsf {Exp}(X)=\mathfrak {p}$
. Since X is zero-dimensional, Lindelöf, and non-compact, it is possible to fix non-empty clopen subsets
$V_n$
of X for
$n\in \omega $
such that
$\bigcup _{n\in \omega }V_n=X$
and
$V_m\cap V_n=\varnothing $
whenever
$m\neq n$
. Also fix an independent family
$\mathcal {A}$
of size
$\kappa $
(see [Reference Kunen8, Exercise III.1.35]), and set
$$ \begin{align*}\mathcal{F}=\{F\subseteq\omega:A_0\cap\cdots\cap A_k\subseteq^\ast F\text{ for some }k\in\omega\text{ and }A_0,\ldots,A_k\in\mathcal{A}\}. \end{align*} $$
Define the space
$X_\kappa $
by taking
$X\cup \{\ast \}$
as the underlying set, where
$\ast \notin X$
simply denotes an extra point, and by declaring
$U\subseteq X\cup \{\ast \}$
open exactly when one of the following conditions holds:
-
• U is an open subset of X,
-
•
$U=\{\ast \}\cup U'$
for some open subset
$U'$
of X such that
$\bigcup \{V_n:n\in F\}\subseteq U'$
for some
$F\in \mathcal {F}$
.
It is straightforward to check that this topology is regular, hence zero-dimensional by [Reference Engelking3, Corollary 6.2.8], hence Tychonoff. Notice that every point of X is non-isolated in
$X_\kappa $
because X is crowded, while
$\ast $
is non-isolated in
$X_\kappa $
because
$\varnothing \notin \mathcal {F}$
. This means that
$X_\kappa $
is crowded.
Claim.
$\chi (\ast ,X_\kappa )=\kappa $
.
Proof It is clear that the open sets of the form
$$ \begin{align*}\{\ast\}\cup\bigcup\{V_n:n\in (A_0\cap\cdots\cap A_k)\setminus\ell\}, \end{align*} $$
where
$k,\ell \in \omega $
and
$A_0,\ldots ,A_k\in \mathcal {A}$
, constitute a local base for
$X_\kappa $
at
$\ast $
. Therefore
$\chi (\ast ,X_\kappa )\leq \kappa $
. Now assume, in order to get a contradiction, that there exists a local base
$\mathcal {B}$
for
$X_\kappa $
at
$\ast $
such that
$|\mathcal {B}|<\kappa $
, and assume without loss of generality that every element of
$\mathcal {B}$
is in the form described above. Since
$|\mathcal {A}|=\kappa $
, we can fix
$A\in \mathcal {A}$
that does not appear in any of these descriptions of the elements of
$\mathcal {B}$
. Since
$\mathcal {B}$
is a local base, there must be
$k,\ell \in \omega $
and
$A_0,\dots ,A_k\in \mathcal {A}\setminus \{A\}$
such that
$$ \begin{align*}\{\ast\}\cup\bigcup\{V_n:n\in (A_0\cap\cdots\cap A_k)\setminus\ell\}\subseteq \{\ast\}\cup\bigcup\{V_n:n\in A\}. \end{align*} $$
It follows that
$(A_0\cap \cdots \cap A_k)\setminus \ell \subseteq A$
, which contradicts the fact that
$\mathcal {A}$
is an independent family.
Using the above claim and the fact that
$\mathsf {w}(X)\leq \mathsf {Exp}(X)=\mathfrak {p}\leq \kappa $
, one sees that
$\mathsf {w}(X_\kappa )=\kappa $
, which clearly implies
$\mathsf {Exp}(X_\kappa )\geq \kappa $
. To conclude the proof, simply apply Proposition 3.4 with
$X_0=X$
and
$X_1=\{\ast \}$
.
7. The pseudocharacter of countable subsets of products
The main purpose of this section is to give an upper bound for
$\mathsf {Exp}(X)$
when X is a countable space (see Corollary 7.3). This will follow from a general analysis of the pseudocharacter of countable subsets of products of first-countable spaces. As a by-product, we will also obtain improved versions of some of the examples given by Theorem 6.1 (see Corollary 7.6).
Lemma 7.1. Let Z be a first-countable space. Then
$\psi (X,Z)\leq \mathfrak {d}$
for every countable subset X of Z.
Proof Pick a countable subset X of Z. The desired result is trivial if X is empty, so assume that X is non-empty and let
$X=\{x_n:n\in \omega \}$
be an enumeration. Given
$n\in \omega $
, fix a local base
$\{U_{n,m}:m\in \omega \}$
for Z at
$x_n$
such that
$U_{n,0}\supseteq U_{n,1}\supseteq \dots $
. Also fix
$\{f_\xi :\xi <\mathfrak {d}\}\subseteq \omega ^\omega $
such that for every
$f:\omega \longrightarrow \omega $
there exists
$\xi <\mathfrak {d}$
such that
$f(n)\leq f_\xi (n)$
for every
$n\in \omega $
. Set
$$ \begin{align*}U_\xi=\bigcup_{n\in\omega}U_{n,f_\xi(n)} \end{align*} $$
for
$\xi <\mathfrak {d}$
, and observe that each
$U_\xi $
is an open subset of Z. It is straightforward to verify that
$X=\bigcap _{\xi <\mathfrak {d}}U_\xi $
, which concludes the proof.
Theorem 7.2. Let
$\kappa $
be an infinite cardinal, and let
$Z_\xi $
for
$\xi \in \kappa $
be first-countable spaces. Set
$Z=\prod _{\xi \in \kappa }Z_\xi $
. Then
$\psi (X,Z)\leq \mathsf {max}\{\mathfrak {d},\kappa \}$
for every countable subset X of Z.
Proof We will proceed by transfinite induction on
$\kappa $
. The case
$\kappa =\omega $
is given by Lemma 7.1. Now assume that
$\kappa $
is an uncountable cardinal and that the desired result holds for all infinite cardinals below
$\kappa $
. Pick a countable subset X of Z. Fix
$\Omega _\xi \subseteq \kappa $
for
$\xi \in \kappa $
so that the following conditions are satisfied:
-
(1)
$x\upharpoonright \Omega _0\neq x'\upharpoonright \Omega _0$
whenever
$x,x'\in X$
are distinct, -
(2)
$\Omega _\xi \subseteq \Omega _{\xi '}$
whenever
$\xi \leq \xi '$
, -
(3)
$|\Omega _\xi |<\kappa $
for each
$\xi $
, -
(4)
$\bigcup _{\xi \in \kappa }\Omega _\xi =\kappa $
.
Given
$\Omega \subseteq \kappa $
, set
$Z(\Omega )=\prod _{\xi \in \Omega }Z_\xi $
. Given
$\xi \in \kappa $
, denote by
$\pi _\xi :Z\longrightarrow Z(\Omega _\xi )$
the natural projection, which is of course obtained by setting
$\pi _\xi (z)=z\upharpoonright \Omega _\xi $
. By condition
$(3)$
and the inductive hypothesis, for every
$\xi \in \kappa $
we can fix a collection
$\mathcal {O}_\xi $
consisting of open subsets of
$Z(\Omega _\xi )$
such that
$|\mathcal {O}_\xi |\leq \mathsf {max}\{\mathfrak {d},\kappa \}$
and
$\bigcap \mathcal {O}_\xi =\pi _\xi [X]$
. Now define
$$ \begin{align*}\mathcal{O}^{\prime}_\xi=\{U\times Z(\kappa\setminus\Omega_\xi):U\in\mathcal{O}_\xi\} \end{align*} $$
for
$\xi \in \kappa $
, where we identify Z with
$Z(\Omega _\xi )\times Z(\kappa \setminus \Omega _\xi )$
in the obvious way. Observe that each
$|\mathcal {O}^{\prime }_\xi |\leq \mathsf {max}\{\mathfrak {d},\kappa \}$
. Therefore, the following claim will conclude the proof.
Claim.
$\bigcap _{\xi \in \kappa }\bigcap \mathcal {O}^{\prime }_\xi =X$
.
Proof The inclusion
$\supseteq $
is trivial. In order to prove the inclusion
$\subseteq $
, pick z that belongs to the left-hand side. It is easy to realize that
$\pi _\xi (z)\in \pi _\xi [X]$
for each
$\xi $
. So there exist
$x_\xi \in X$
for
$\xi \in \kappa $
such that each
$\pi _\xi (x_\xi )=\pi _\xi (z)$
. However, using conditions
$(1)$
and
$(2)$
, one sees that there must be
$x\in X$
such that each
$x_\xi =x$
. Finally, by condition
$(4)$
, it is clear that
$z=x\in X$
, as desired.
Corollary 7.3. Let X be a countable space. Then
$\mathsf {Exp}(X)\leq \mathfrak {c}$
.
Proof Observe that
$\mathsf {w}(X)\leq \mathfrak {c}$
because X is countable. Therefore, since X is zero-dimensional by [Reference Engelking3, Corollary 6.2.8], we can assume that X is a subspace of
$2^{\mathfrak {c}}$
by [Reference Engelking3, Theorem 6.2.16]. Observe that
$\mathsf {cl}(X)$
is a compactification of X, where
$\mathsf {cl}$
denotes closure in
$2^{\mathfrak {c}}$
. Therefore, by Theorem 7.2, there exist open subsets
$U_\xi $
of
$2^{\mathfrak {c}}$
for
$\xi \in \mathfrak {c}$
such that
$X=\bigcap _{\xi \in \mathfrak {c}}U_\xi $
. It follows that
$$ \begin{align*}\mathsf{cl}(X)\setminus X=\bigcup_{\xi\in\mathfrak{c}}(\mathsf{cl}(X)\setminus U_\xi), \end{align*} $$
which shows that
$\mathsf {kc}^\ast (X)\leq \mathfrak {c}$
by Proposition 2.1. In conclusion, an application of Theorem 3.2 yields the desired result.
Corollary 7.4. Let
$\kappa \geq \mathfrak {d}$
be a cardinal, and let
$Z_\xi $
for
$\xi \in \kappa $
be first-countable spaces such that each
$|Z_\xi |\geq 2$
. Set
$Z=\prod _{\xi \in \kappa }Z_\xi $
. Then
$\psi (X,Z)=\kappa $
for every non-empty countable subset X of Z.
Proof Pick a non-empty countable subset X of Z. First observe that
$\psi (X,Z)\leq \kappa $
by Theorem 7.2. Now assume, in order to get a contradiction, that
$\psi (X,Z)<\kappa $
. Let
$\mathcal {O}$
be a collection of open subsets of Z such that
$|\mathcal {O}|<\kappa $
and
$\bigcap \mathcal {O}=X$
. Fix
$x\in X$
, then define
$$ \begin{align*}\mathcal{O}'=\mathcal{O}\cup\{Z\setminus\{z\}:z\in X\setminus\{x\}\}. \end{align*} $$
It is clear that
$|\mathcal {O}'|<\kappa $
and
$\bigcap \mathcal {O}'=\{x\}$
. Since each
$|Z_\xi |\geq 2$
, this contradicts the fact that
$2^\kappa $
has pseudocharacter
$\kappa $
at each point (see [Reference Juhász6, 5.3(b)]).
Before stating the next two results, we remind the reader that a product of at most
$\mathfrak {c}$
separable spaces is separable (see [Reference Engelking3, Corollary 2.3.16]).
Corollary 7.5. Let
$\kappa $
be a cardinal such that
$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$
, and let
$Z_\xi $
for
$\xi \in \kappa $
be compact separable first-countable spaces such that each
$|Z_\xi |\geq 2$
. Set
$Z=\prod _{\xi \in \kappa }Z_\xi $
. If X is a countable dense subspace of Z then
$\mathsf {Exp}(X)=\kappa $
.
Proof Pick a countable dense subspace X of Z. The desired result follows from Theorem 3.2 and Corollary 7.4, since Z is a compactification of X.
Corollary 7.6. Let
$\kappa $
be a cardinal such that
$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$
. Then there exists a countable
$($
crowded
$)$
topological group X such that
$\mathsf {Exp}(X)=\kappa $
.
Proof Pick a countable dense subset D of
$2^\kappa $
, and let X be the subgroup of
$2^\kappa $
generated by D. It follows from Corollary 7.5 that
$\mathsf {Exp}(X)=\kappa $
.
8. Scattered spaces
The purpose of this section is to show that the realcompactness number of a countable space is essentially determined by its perfect kernel (see Corollary 8.4). In order to achieve this, we will obtain a bound for the pseudocharacter of a scattered space in any compactification (see Theorem 8.2).
We begin with a technical lemma, which was inspired by [Reference Engelking3, Theorem 3.9.2]. Recall that an extension of a space X is a space Z in which X is dense.
Lemma 8.1. Let X be a space, let
$\kappa $
be an infinite cardinal, and let
$\mathcal {C}_\xi $
for
$\xi \in \kappa $
be open covers of X. Assume that
$\bigcap \mathcal {D}\neq \varnothing $
whenever
$\mathcal {D}$
satisfies the following requirements
$:$
-
(1)
$\mathcal {D}$
consists of closed subsets of X, -
(2)
$D_0\cap \cdots \cap D_n\neq \varnothing $
whenever
$n\in \omega $
and
$D_0,\ldots ,D_n\in \mathcal {D}$
, -
(3)
$\forall \xi \in \kappa \,\exists D\in \mathcal {D}\,\exists U\in \mathcal {C}_\xi \, (D\subseteq U)$
.
Then
$\psi (X,Z)\leq \kappa $
for every extension Z of X.
Proof Pick an extension Z of X. Define
$$ \begin{align*}U_\xi=\bigcup\{U\text{ open in }Z:U\cap X\in\mathcal{C}_\xi\} \end{align*} $$
for
$\xi \in \kappa $
, and observe that each
$U_\xi $
is open in Z. Therefore, the following claim will conclude the proof.
Claim.
$X=\bigcap _{\xi \in \kappa }U_\xi $
.
Proof The inclusion
$\subseteq $
is clear, since each
$\mathcal {C}_\xi $
is an open cover of X. In order to prove the inclusion
$\supseteq $
, pick
$x\in \bigcap _{\xi \in \kappa }U_\xi $
. Define
$$ \begin{align*}\mathcal{D}=\{\mathsf{cl}(U)\cap X:U\text{ is an open neighborhood of }x\text{ in }Z\}, \end{align*} $$
where
$\mathsf {cl}$
denotes closure in Z, and observe that
$\bigcap \mathcal {D}\subseteq \{x\}$
. It is obvious that condition
$(1)$
holds. Using the fact that X is dense in Z, one sees that condition
$(2)$
holds. Finally, using the assumption that
$x\in \bigcap _{\xi \in \kappa }U_\xi $
, it is straightforward to verify that condition
$(3)$
holds. Therefore
$\bigcap \mathcal {D}\neq \varnothing $
, which implies
$\bigcap \mathcal {D}=\{x\}$
. Since clearly
$\bigcap \mathcal {D}\subseteq X$
, it follows that
$x\in X$
.
Given a scattered space X and
$S\subseteq X$
, define the Cantor–Bendixson rank of S as
$$ \begin{align*}\mathsf{rank}(S)=\mathsf{min}\{\xi:X^{(\xi)}\cap S=\varnothing\}. \end{align*} $$
When
$S=\{x\}$
, we will simply write
$\mathsf {rank}(x)$
to mean
$\mathsf {rank}(S)$
.
Theorem 8.2. Let X be a scattered space, and let
$\kappa =\mathsf {w}(X)$
. Then
$\psi (X,Z)\leq \kappa $
for every extension Z of X.
Proof Assume without loss of generality that X is non-empty. Define a subset S of X to be focused if there exists a unique
$x\in S$
such that
$\mathsf {rank}(x)=\mathsf {rank}(S)$
. Notice that
$\mathsf {rank}(x)$
is a successor ordinal for every
$x\in X$
. Given
$x\in X$
such that
$\mathsf {rank}(x)=\xi +1$
, set
$$ \begin{align*}U_x=\big(X\setminus X^{(\xi)}\big)\cup\{x\}. \end{align*} $$
It is not hard to realize that each
$U_x$
is a focused open neighborhood of x. Furthermore, it is clear that every neighborhood of x contained in
$U_x$
will still be focused. Using this fact, it is possible to fix a base
$\mathcal {B}$
for X of size at most
$\kappa $
such that
$\mathcal {B}$
consists of focused open sets. Then define
$$ \begin{align*}\mathcal{C}_0=\{U\in\mathcal{B}:\mathsf{cl}(U)\subseteq V\text{ for some }V\in\mathcal{B}\text{ with }\mathsf{rank}(V)=\mathsf{rank}(U)\}. \end{align*} $$
Claim 1.
$\mathcal {C}_0$
is a base for X consisting of focused open sets.
Proof Pick
$x\in X$
and an open neighborhood U of x. Since
$\mathcal {B}$
is a base, there exists
$V\in \mathcal {B}$
such that
$x\in V\subseteq U\cap U_x$
. By regularity, there exists
$U'\in \mathcal {B}$
such that
$x\in U'\subseteq \mathsf {cl}(U')\subseteq V$
. It is clear that
$U'\in \mathcal {C}_0$
, as desired.
Given
$U_0,U_1,U_2,U_3\in \mathcal {B}$
such that
$\mathsf {cl}(U_3)\subseteq U_2\subseteq U_1\subseteq \mathsf {cl}(U_1)\subseteq U_0$
, define
$$ \begin{align*}\mathcal{C}_{U_0,U_1,U_2,U_3}=\{U_2\}\cup\{V\in\mathcal{C}_0:V\subseteq U_0\setminus\mathsf{cl}(U_3)\}\cup\{X\setminus\mathsf{cl}(U_1)\}, \end{align*} $$
and observe that each
$\mathcal {C}_{U_0,U_1,U_2,U_3}$
is an open cover of X. Notice that there exists at least one such cover because X is non-empty, and let
$\mathcal {C}_\xi $
for
$\xi \in \kappa \setminus \{0\}$
enumerate them all (possibly with repetitions). By Lemma 8.1, it will be enough to show that
$\bigcap \mathcal {D}\neq \varnothing $
whenever
$\mathcal {D}$
satisfies conditions
$(1)$
–
$(3)$
. So pick such a
$\mathcal {D}$
.
Without loss of generality, assume that
$\mathcal {D}$
is closed under finite intersections. Set
$\xi =\mathsf {min}\{\mathsf {rank}(D):D\in \mathcal {D}\}$
. By considering
$\mathcal {D}'=\{D'\cap D:D\in \mathcal {D}\}$
instead of
$\mathcal {D}$
, where
$D'\in \mathcal {D}$
is a fixed element of rank
$\xi $
, we can also assume that
$\mathsf {rank}(D)=\xi $
for every
$D\in \mathcal {D}$
. By condition
$(3)$
applied to the cover
$\mathcal {C}_0$
, we can define
$$ \begin{align*}\xi'=\mathsf{min}\{\mathsf{rank}(U):U\in\mathcal{C}_0\text{ and }D\subseteq U\text{ for some }D\in\mathcal{D}\}. \end{align*} $$
Claim 2.
$\xi '=\xi $
.
Proof It is easy to realize that
$\xi \leq \xi '$
. Now assume, in order to get a contradiction, that
$\xi <\xi '$
. Pick
$U_1\in \mathcal {C}_0$
and
$D\in \mathcal {D}$
such that
$\mathsf {rank}(U_1)=\xi '$
and
$D\subseteq U_1$
. By the definition of
$\mathcal {C}_0$
, we can also fix
$U_0\in \mathcal {B}$
such that
$\mathsf {cl}(U_1)\subseteq U_0$
and
$\mathsf {rank}(U_0)=\mathsf {rank}(U_1)$
. Since
$\mathcal {C}_0$
consists of focused sets, there exists a unique
$x\in U_1$
such that
$\mathsf {rank}(x)=\xi '$
. Notice that x is also the unique element of
$U_0$
whose rank is
$\xi '$
. The fact that
$\mathsf {rank}(D)=\xi <\xi '=\mathsf {rank}(x)$
shows that
$x\notin D$
. Therefore, by Claim 1, it is possible to find
$U_2\in \mathcal {C}_0$
such that
$x\in U_2\subseteq \mathsf {cl}(U_2)\subseteq U_1\setminus D$
. By regularity, we can also fix
$U_3\in \mathcal {B}$
such that
$x\in U_3\subseteq \mathsf {cl}(U_3)\subseteq U_2$
.
By condition
$(3)$
, there exist
$V\in \mathcal {C}_{U_0,U_1,U_2,U_3}$
and
$D'\in \mathcal {D}$
such that
$D'\subseteq V$
. It follows from condition
$(2)$
that
$V\neq U_2$
and
$V\neq X\setminus \mathsf {cl}(U_1)$
, hence
$V\in \mathcal {C}_0$
and
$V\subseteq U_0\setminus \mathsf {cl}(U_3)$
. It follows that
$\mathsf {rank}(V)\geq \xi '$
by minimality, while on the other hand
$\mathsf {rank}(V)\leq \mathsf {rank}(U_0)=\xi '$
. In conclusion, we have shown that
$\mathsf {rank}(V)=\xi '$
. Since
$V\in \mathcal {C}_0$
is a focused set, it follows that there exists
$y\in V$
such that
$\mathsf {rank}(y)=\xi '$
. This is a contradiction, since x is the unique element of
$U_0$
of rank
$\xi '$
.
By Claim 2, we can fix
$U\in \mathcal {C}_0$
and
$D\in \mathcal {D}$
such that
$\mathsf {rank}(U)=\xi $
and
$D\subseteq U$
. Let x be the unique element of U of rank
$\xi $
. Clearly, the following claim will conclude the proof.
Claim 3.
$x\in \bigcap \mathcal {D}$
.
Proof Pick any
$D'\in \mathcal {D}$
. Observe that
$D'\cap D\in \mathcal {D}$
because
$\mathcal {D}$
is closed under finite intersections, hence
$\mathsf {rank}(D'\cap D)=\xi $
. Since
$D\cap D'\subseteq D\subseteq U$
, it follows that
$x\in D\cap D'\subseteq D'$
, as desired.
We remark that the proof of Theorem 8.2 would be more pleasant under the assumption that X is zero-dimensional. However, it is not true that every scattered space is zero-dimensional (see [Reference Solomon11]).
Corollary 8.3. Let X be a Lindelöf scattered space. Then
$$ \begin{align*}\mathsf{Exp}(X)=\mathsf{w}(X). \end{align*} $$
Corollary 8.4. Let X be a hereditarily Lindelöf space. Then
$$ \begin{align*}\mathsf{Exp}(X)=\mathsf{max}\{\mathsf{Exp}(\mathsf{ker}(X)),\mathsf{w}(X)\}. \end{align*} $$
Proof Set
$X_0=\mathsf {ker}(X)$
and
$X_1=X\setminus X_0$
. The inequality
$\geq $
is clear, since
$X_0$
is a closed subspace of X. In order to prove the inequality
$\leq $
, observe that
$X_1$
is a scattered Lindelöf space, hence
$\mathsf {Exp}(X_1)=\mathsf {w}(X_1)$
by Corollary 8.3. The desired result then follows from Proposition 3.4.
9. Open questions
Our first three questions aim at “improving” the examples given in Sections 5 and 6. Recall that a space X is extremally disconnected if the closure of every open subset of X is open.
Question 9.1. For which cardinals
$\kappa $
such that
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$
does there exist an extremally disconnected countable crowded space X such that
$\mathsf {Exp}(X)=\kappa $
?
We do not know whether the assumption that
$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$
in Corollary 7.6 can be weakened to
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$
. This is the content of the next question.
Question 9.2. For which cardinals
$\kappa $
such that
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$
does there exist a countable (crowded) topological group X such that
$\mathsf {Exp}(X)=\kappa $
?
Recall that a space X is homogeneous if for all
$x,y\in X$
there exists a homeomorphism
$h:X\longrightarrow X$
such that
$h(x)=y$
. Since every topological group is clearly homogeneous, the following might be a first step towards answering Question 9.2.
Question 9.3. For which cardinals
$\kappa $
such that
$\mathfrak {p}\leq \kappa \leq \mathfrak {c}$
does there exist a countable (crowded) homogeneous space X such that
$\mathsf {Exp}(X)=\kappa $
?
Finally, it would be desirable to “complete the picture” regarding the results of Section 7. Notice that, by Corollary 7.4, it only remains to answer Question 9.4 in the case when
$\kappa <\mathfrak {d}$
. We remark that we do not have a complete answer even in the case
$\kappa =1$
.
Question 9.4. Assume that
$\kappa $
is a given cardinal. What are the possible values of
$\psi (X,Z)$
when Z is a product of
$\kappa $
first-countable spaces and X is a countable subset of Z?
Funding
This research was funded in whole or in part by the Austrian Science Fund (FWF) grants P 35655 (Claudio Agostini and Andrea Medini), P 35588 (Andrea Medini), and I 5930 (Lyubomyr Zdomskyy). For open access purposes, the authors have applied a CC BY public copyright license to any authors accepted manuscript version arising from this submission.
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