2 Morita’s theorem and locally closed maps

Let Y be a locally compact, $\sigma $ -compact Hausdorff space and $q:Y\to X$ a quotient map with X Hausdorff. Then, we can write $Y=\bigcup _{i\ge 1} Y_i$ where each $Y_i$ is compact and is contained in the interior of its successor. We will say that $\{Y_i: i\ge 1\}$ is a compact decomposition for Y, while the compact sets $\{ q(Y_i): i\ge 1\}$ are a $k_{\omega }$ -decomposition for X. The space X is hemicompact with regard to a $k_{\omega }$ -decompositon; that is, for any compact $K\subseteq X$ , eventually $K\subseteq q(Y_i)$ for some i (see [Reference Franklin and Smith Thomas10, p. 113]).

One of the central results on $k_{\omega }$ -spaces is due to Morita [Reference Morita19, Theorem 4].

As Morita observes, the discrete set of exceptional points is in fact contained in a larger discrete set of points with noncompact fiber. For all other points, the fiber is compact.

If q is not closed, the situation becomes more complicated, and to discuss this the following definition is useful. Let X and Y be topological spaces. A map $q:Y\to X$ is locally closed at a point $x\in X$ if for every closed set $W\subseteq Y$ , $x\in \overline {q(W)}$ implies $x\in q(W)$ [Reference Kuratowski13, Section 13.XIV]. Clearly if q is surjective then q is closed if and only q is locally closed at each point of x. One basic result is as follows.

In general, there seems to be little relation between the compactness of the fiber and “local closedness” of the quotient map on the one hand, and the local compactness of the image on the other. The following example is typical. Let $Y=\bigcup _{n\ge 0} L_n$ where $L_n$ is the line $y=n$ in $ {\mathbb {R}}^2$ , and let $q:Y\to X$ be the projection of Y onto the x-axis, which is a quotient map. Then X is locally compact and Hausdorff but q is nowhere locally closed and $q^{-1}(x)$ has noncompact boundary for every $x\in X$ .

A rather similar definition to “locally closed” was introduced independently by various authors, the earliest of whom seems to have been McDougle [Reference Malyhin17]. Let X and Y be topological spaces. A surjective map $f:Y\to X$ is pseudo-open at $x\in X$ if whenever U is an open subset of Y containing $f^{-1}(x)$ , $f(U)$ is a neighborhood of x. If f is pseudo-open at every point of X then f is said to be pseudo-open. A pseudo-open map is easily seen to be a quotient map, and if f is locally closed at x then f is pseudo-open at x. The example above, with q open but nowhere locally closed, shows that the converse is not true. We will see, however, that the two conditions are equivalent if the quotient map q is pure.

The next theorem was essentially proved in [Reference Arkhangel’skii2, Theorems 3.3 and 3.4] (see also [Reference Arkhangel’skii1]) but our statement of it is somewhat different, so we include the short proof here.

For a quotient map $q: Y\to X$ , let $H_q$ be the set of points in X at which q is pseudo-open. Theorem 1.3 shows that if Y is a locally compact $\sigma $ -compact Hausdorff space and X is Hausdorff then $H_q$ is the set of k-points in X.

Among the k-points, the points x of local compactness in a $k_{\omega }$ -space (and more generally) can be characterized as those which satisfy the biquotient property: every open cover $\mathcal {U}$ of $q^{-1}(x)$ has a finite subset $\{U_1, \ldots , U_n\}\subseteq \mathcal {U}$ such that $\{ q(U_1), \ldots , q(U_n)\}$ covers a neighborhood of x (see [Reference Michael18, Reference Stone23]). While the characterizations given by the biquotient property and by pseudo-openness are elegant and versatile, they do not seem to lead to a structure theory for general $k_{\omega }$ -spaces along the lines of Morita’s theorem. Something is needed to bring the ideas together.

3 Pure quotient maps

In this section, we introduce pure quotient maps. They are weaker than closed quotient maps in one important aspect, but with a compensatory strengthening in another direction; and while they retain some of the nice properties of closed quotient maps, they are much more general. We begin with a preparatory lemma.

Condition (ii) is equivalent to requiring that if $d_{\alpha }\to \infty $ then the net $q^{-1}(q(d_{\alpha }))$ converges to infinity in the Fell topology. Lemma 3.1 shows that condition (ii) is automatically satisfied whenever q is a closed quotient map.

On the other hand, condition (i) implies that for every $x\in X$ , the interior of $q^{-1}(x)$ is either empty or consists of an isolated point. In fact, condition (i) is equivalent to this when Y is second countable. To see this, let $D_0$ be the (countable) set of isolated points in Y and fix a countable base $\{U_i\}_{i\ge 1}$ for $Y\setminus \overline {D_0}$ . Inductively chose a point from each $U_i$ which does not belong to any fiber with nonempty interior or any fiber previously chosen. This is possible because $U_i$ contains no isolated point and hence is an uncountable Baire space, and every fiber intersects $U_i$ in a closed set with empty interior. Let $D_1$ be the set thus obtained, and set $D=D_0\cup D_1$ . Then D is dense in Y and q restricted to D is injective.

Here is an alternative description of pure quotient maps that will be useful later.

Thus, if q is a pure quotient map and $Y=\bigcup _{i\ge 1} Y_i$ is a compact decomposition of Y, then for each $Y_i$ we may find $Y_{d(i)}$ with $d(i)\ge i$ such that $D_i\subseteq Y_{d(i)}$ .

The usefulness of pure quotient maps appears in the following converse to Proposition 2.2 (note that because a fiber of a pure quotient map has at most a singleton as its interior, the fiber is compact if and only if its boundary is compact).

Pure quotient maps are not uncommon. Clearly, every locally compact $\sigma $ -compact Hausdorff space X admits the trivial pure quotient, taking $Y=X$ and q as the identity map, and we now show that every $k_{\omega }$ -space X admits a pure quotient. Specifically, we show that if Y is a locally compact, $\sigma $ -compact Hausdorff space, and $q:Y\to X$ a quotient map with X Hausdorff, then q can be “purified”: that is, a locally compact $\sigma $ -compact space $Y_*$ can be derived from Y such that the quotient map $q_*:Y_*\to X$ is pure (and induces the same topology on X as q).

For a topological space X, let $\operatorname {Cl}(X)$ be the hyperspace of closed subsets of X with the Fell topology. A base for the Fell topology consists of the family of all sets

$$ \begin{align*}U(K, \Phi)=\{ S\in \operatorname{Cl}(X): S\cap K=\emptyset,\ S\cap O\ne\emptyset,\ O\in \Phi\},\end{align*} $$

where K is a compact subset of X and $\Phi $ is a finite family of open subsets of X. Then $\operatorname {Cl}(X)$ is a compact space [Reference Fell9, Lemma 1], and is Hausdorff if X is locally compact [Reference Fell9, Theorem 1]. The next lemma is presumably standard.

It follows from Lemma 3.5 that, in the context of the lemma, we may consistently define a map $q_*: Y_*\to X$ by $q_*(F)=q(y) (y\in F)$ .

Note that $q_*^{-1}(x)$ is the set consisting of $q^{-1}(x)$ and a family of closed subsets of the boundary of $q^{-1}(x)$ , obtained as limits of nets of fibers; see Lemma 3.5. (Thus in the example after Proposition 2.2, D is homeomorphic to ${\mathbb {R}}$ and is closed in $\operatorname {Cl}(Y)\setminus \{\emptyset \}$ ). If the boundary of $q^{-1}(x)$ is compact then $q_*^{-1}(x)$ is also compact because $Y_*$ is the closure of D in $\operatorname {Cl}(Y)\setminus \{\emptyset \}$ . On the other hand, if the boundary of $q^{-1}(x)$ is noncompact then a net of its closed subsets might converge to $\emptyset $ in the Fell topology, so $q_*^{-1}(x)$ might be compact or noncompact.

Apart from the fact that its fibers are more likely to be compact (or to have compact boundary) than those of q, a further advantage that $q_*$ has over q is that it is more likely to be locally closed.

In particular, if q is closed then $q_*$ is closed. Note, however, that in this case if the boundary of $q^{-1}(x)$ is noncompact then x does not have a compact neighborhood in X by Theorem 2.1, and hence $q_*^{-1}(x)$ is noncompact by Proposition 2.2 applied to $q_*$ .

Proposition 3.7 raises the question of whether $q_{**}$ might be locally closed at even more points of X than $q_*$ is, but we will see in the next section that $q_*$ already reaches the limit.

4 An extension of Morita’s theorem

In this section, we use pure quotient maps to extend Morita’s theorem to general $k_{\omega }$ -spaces.

Given a quotient map q from a locally compact $\sigma $ -compact Hausdorff space Y onto a Hausdorff space X, we partition X into three sets as follows:

(i) $L_q=\{x\in X: q$ is locally closed at x and $q^{-1}(x)$ has compact boundary};

(ii) $F_q=\{ x\in X: q$ is locally closed at x and $q^{-1}(x)$ has noncompact boundary}; and (iii) $N_q=\{ x\in X: q$ is not locally closed at x}.

We saw in Proposition 2.2 that if $x\in L_q$ then x has a compact neighborhood in X, and in Theorem 3.3 that if q is a pure quotient map then $L_q$ is precisely the set of points in X which have a compact neighborhood. We now show that if q is a pure quotient map, the sets $F_q$ and $N_q$ can similarly be characterized in terms of the topology of X.

Proof We have seen that if q is locally closed at x then it is pseudo-open at x, and hence x is a k-point by Theorem 2.3.

Conversely, suppose that q is not locally closed at x, and let N be a closed subset of Y disjoint from $q^{-1}(x)$ with $x\in \overline {q(N)}$ . Let W be a closed subset of Y disjoint from $q^{-1}(x)$ with N in its interior. Let D be the dense subset of Y from the pure quotient property, and set $M=W\cap D$ and $E=q(M)$ . Then $\overline M\supseteq N$ , so $x\in \overline {E}\setminus E$ . Let $Y=\bigcup _i Y_i$ be a compact decomposition for Y and set $K_i=q(Y_i)$ . Then for any i, $q^{-1}(K_i\cap E)=\{ d\in M: q(d)\in K_i\}$ so $q^{-1}(K_i\cap E)\subseteq Y_{d(i)}$ (in the terminology introduced after Lemma 3.2). Thus

$$ \begin{align*}K_i\cap E\subseteq q(Y_{d(i)}\cap M)\subseteq q(Y_{d(i)}\cap \overline M)\subseteq q(Y_{d(i)}\cap W)\end{align*} $$

and this latter set is compact and does not contain x. Since every compact subset of X is contained in some $K_i$ by hemicompactness, it follows that there does not exist compact K with $x\in \overline {E\cap K}$ . Hence x is not a k-point.▪

Thus if q is a pure quotient map, q is locally closed at x if and only if q is pseudo-open at x. The following corollary is immediate from Theorem 4.1.

In particular, for any quotient map with domain a locally compact, $\sigma $ -compact Hausdorff space and Hausdorff range, $N_{q_{**}}=N_{q_*}$ , answering the question raised in Section 3. It follows from Corollary 4.2 that for a $k_{\omega }$ -space X, the sets $L_{q_*}$ , $F_{q_*}$ , and $N_{q_*}$ are independent of the original quotient map q, and in future we may use this notation even when there is no particular original quotient map in mind.

The next corollary is also immediate from Theorem 4.1 (applied to $q_*$ ) and Theorem 2.3 (applied to q). Recall that $H_q$ denotes the set of points of X at which q is pseudo-open.

We turn now to consider the topology of the sets $L_{q}$ , $F_{q}$ , and $N_{q}$ when q is pure (in which case $H_q=L_q\cup F_q$ by Corollary 4.3). It follows from Corollary 4.2 that $L_q$ is open, and hence that $F_q\cup N_q$ is closed in X.

Recall that a point x in a Tychonoff space X is a P-point if the intersection of any countable family of neighborhoods of x is a neighborhood of x. If every point is a P-point then X is a P-space. If U is an open subset of X and x is a P-point in U then x is P-point in X. Conversely, a P-point in X is a P-point in any subspace of X, and hence the set of P-points of X forms a P-space. If X is a compact P-space then X is finite [Reference Gillman and Jerison11, 4K]. The next theorem shows that $F_q$ is contained in the set of P-points of $F_q\cup N_q$ . For this, we need the following observation.

Let Y be a locally compact $\sigma $ -compact Hausdorff space and $q: Y\to X$ a quotient map with X Hausdorff. Then q extends to a continuous map $\overline q: \beta Y\to \beta X$ , and $\overline q$ is continuous and surjective from a compact Hausdorff space, and is therefore also a quotient map. If $x\in X$ and q is not locally closed at x, then there is a closed set $N\subseteq Y$ disjoint from $q^{-1}(x)$ such that $x\in \overline {q(N)}$ . Hence, there is a point $y\in \overline {N}^{\beta Y}$ such that $\overline q(y)=x$ .

Theorem 4.5 Let Y be a locally compact $\sigma $ -compact Hausdorff space and $q: Y\to X$ a pure quotient map with X Hausdorff.

(i) If $x\in X$ is a P-point then q is locally closed at x.

(ii) Let W be the set of P-points in $ F_q\cup N_q$ and V the interior of $F_q\cup N_q$ in X. Then

$$ \begin{align*}V\cap W\subseteq F_q\subseteq W.\end{align*} $$

(iii) $F_q$ is a P-space and has empty interior in X.

Proof (i) (This does not require q to be pure). Let C be a closed subset of Y disjoint from $q^{-1}(x)$ . Then

$$ \begin{align*}C=\bigcup_{n\ge 1} C\cap K_n,\end{align*} $$

where $Y=\bigcup _{n\ge 1} K_n$ is a compact decomposition for Y, so $q(C)=\bigcup _{n\ge 1} q(C\cap K_n)$ . Since $q(C\cap K_n)$ is compact, hence closed in X, the P-point x does not lie in $\overline {q(C)}$ . Thus q is locally closed at x.

(ii) It follows from (i) that $F_q\supseteq V\cap W$ . Now suppose that $x_0\in F_q\cup N_q$ is not a P-point of $F_q\cup N_q$ . Let $\{E_i: i\ge 1\}$ be a countable family of closed sets in $F_q\cup N_q\setminus \{ x_0\}$ with $x_0\in \overline {\bigcup _i E_i}$ . Fix i, and let $U_i$ be a closed neighborhood of $x_0$ in X that does not meet $E_i$ . For each $x\in E_i$ choose $y\in q^{-1}(x)$ with $y\notin Y_{i+1}$ if this is possible (as it is for each $x\in F_q$ since $q^{-1}(x)$ is noncompact). Otherwise, by the observation preceding Lemma 4.4, there exists $y\in \beta Y\setminus Y$ such that $\overline q(y)=x$ . By Lemma 4.4, there is a zero set Z in Y with $y\in \overline Z^{\beta Y}$ and $Z\cap q^{-1}(U_i)$ empty, and we may also arrange that $Z\cap Y_{i+1}$ is empty.

Let $N_i'$ be the union of the collection of elements y and zero sets Z thus obtained, and let $N_i$ be the closure of $N_i'$ in Y. Then $N_i$ does not meet $Y_i$ since $Y_i$ is contained in the interior of $Y_{i+1}$ , and $\overline {q(N_i)}\supseteq E_i$ , but $q(N_i)\cap U_i$ is empty (so in particular $x_0\notin q(N_i)$ ). Set $N=\bigcup N_i$ . Then N does not meet $q^{-1}(x_0)$ , and N is closed in Y because, for any $Y_j$ , only finitely many of the sets $N_i$ meet $Y_j$ , so $N\cap Y_j$ is closed. But $\overline {q(N)}\supseteq \bigcup _i E_i$ , so $x_0\in \overline {q(N)}\setminus q(N)$ . Hence, $x_0\in N_q$ , so $F_q\subseteq W$ .

(iii) $F_q$ is a P-space by (ii). Let U be the interior of $F_q$ in $F_q\cup N_q$ . Then $O:=q^{-1}(U)$ is open in Y, hence a locally compact Hausdorff space, and $q|_O$ is a quotient map. Thus U is a k-space, so a subset of U is closed if and only if its intersection with every compact subset of U is closed. But U is P-space, so compact subsets are finite [Reference Gillman and Jerison11, 4K]. Thus, every subset of U is closed, so U is discrete. Hence, each point in U lies in the closure of $L_q$ (for otherwise it would be an isolated point of X and would belong to $L_q$ itself). Thus, $F_q$ has empty interior in X.▪

Theorem 4.5 is an extension of Morita’s theorem (1.1) because if q is a closed quotient map then $q_*$ is closed (Proposition 3.7), so $N_q$ and $N_{q_*}$ are both empty. Furthermore $F_q=F_{q_*}$ by the remark after Proposition 3.7, so $F_{q}$ is closed, and thus $\sigma $ -compact. Being a P-space by Theorem 4.5, $F_{q}$ must be a countable discrete set in the closure of $L_q$ , and thus we recover Morita’s theorem.

In some cases, we can go further. Recall that a topological space X has countable tightness at $x\in X$ if for each $A\subseteq X$ with $x\in \overline A$ , there is a countable subset $B\subseteq A$ with $x\in \overline B$ ; and X is countably tight if it is has countable tightness at every point. Clearly every metric space is countably tight.

Corollary 4.6 seems to be new even for quotients of locally compact subsets of ${\mathbb {R}}^n$ .

We have just seen that if X is countably tight then there are no points of $F_q$ in the interior of $N_q\cup F_q$ , but if X is not countably tight, such points may occur. For example, let X be the quotient of $\omega _1+1\times S_{\omega }$ (where $S_{\omega }$ is the Arkhangel’skii–Franklin space [Reference Arkhangel’skii and Franklin6] obtained by identifying $\{\omega _1\}\times S_{\omega }$ to a point $x_0$ . Then $L_q$ is empty and $F_q=\{x_0\}$ .

Not every P-point of $F_q\cup N_q$ need belong to $F_q$ . For example, let $Y={\mathbb {R}}$ and for $n\ge 1$ , identify the pairs of points n and $1/n$ . Let X be the resulting quotient space and $q:Y\to X$ the quotient map, which is obviously pure. Then $F_q\cup N_q$ consists of the singleton $\{ q(0)\}$ , but $F_q$ is empty. The following example, however, has the satisfying feature that $F_q$ consists precisely of the P-points in $F_q\cup N_q$ .

Let $Y=\bigcup _i Z_i$ be the disjoint union of countably many copies of Z. Set $W=\bigcup _i V_i$ , and for each i let $\phi _i$ be a homeomorphism from V to $V_i$ . Let O be the complement of W in Y. Let $\sim $ be the equivalence relation defined on points $y\in Y$ by $y\sim y'$ if and only if either (i) $y=y'$ ; or (ii) $y, y'\in W$ and there exists $v\in V$ such that $\phi _i(v)=y$ and $\phi _j(v)=y'$ for some i and j. Set $X=Y/\sim $ and let q be the quotient map. Then q is pure because V has empty interior in Z. Note that $q(W)$ is homeomorphic to V, and that $q|_{V_i}$ is a homeomorphism from $V_i$ onto $q(W)$ for each i. For ease of notation, we will identify $q(W)$ with V, so that $\phi _i: V\to V_i$ is the inverse of $q|_{V_i}$ .

Clearly $q|_O$ is a homeomorphism from O onto $q(O)$ , so if $x\in q(O)$ then x is a point of local compactness. Now let $x\in V$ and suppose first that x is not a P-point in V. Let $\{N_i\}$ be a countable family of closed subsets of V such that $x\notin N_i$ for all i but $x\in \overline {\bigcup _i N_i}$ . Set $N=\bigcup _i \phi _i(N_i)$ . Then N is a closed subset of Y disjoint from $q^{-1}(x)$ , but $x\in \overline {q(N)}$ . Hence $x\in N_q$ .

Now suppose that that x is a P-point in V. Let N be a closed subset of Y not meeting $q^{-1}(x)$ . Then the sets $q(N\cap V_i)$ are a countable family of closed set in V (since $q|_{V_i}$ is a homeomorphism), and they each miss x, so $M:=\overline {\bigcup _i q(N\cap V_i)}$ also misses x. Set $N'=N\cup \bigcup _i \phi _i(M)$ . Then $N'$ is a closed saturated set in Y containing N and missing $q^{-1}(x)$ , so $x\notin \overline {q(N)}$ . Hence $x\in F_q$ .

Thus in Example 4.7, $N_q$ is closed in X if and only if every P-point in V is an isolated point of V. If V has a nonisolated P-point, for example, if $V=\omega _1+1$ (and Z is the cone over V), then $N_q$ is not closed. If $Z= \beta {\mathbb {N}}$ and $V=\beta {\mathbb {N}}\setminus {\mathbb {N}}$ , then it depends on the set theoretic axioms adopted as to whether $F_q$ is empty or dense in $F_q\cup N_q$ .

5 Arkhangelskii’s question and Fréchet-Urysohn spaces

Finally, we turn to consider first countable and Fréchet-Urysohn spaces and Arkhangel’skii’s question (for a vast amount of information on this whole area, see [Reference Siwiec21]).

As mentioned in the Introduction, a point x in a topological space X is a Fréchet-Urysohn point if whenever $E\subseteq X$ with $x\in \overline E$ there is a sequence $(v_n)$ in E with $v_n\to x$ . Clearly, every point of first countability is a Fréchet-Urysohn point. If every point is a Fréchet-Urysohn point then X is a Fréchet-Urysohn space. A Fréchet-Urysohn point in a $k_{\omega }$ -space is easily seen to be a a k-point (Proposition 5.1(ii)) and thus belongs either to $L_{q_*}$ or to $F_{q_*}$ . Ordman showed that a point of first countability must belong to $L_{q_*}$ (see [Reference Franklin and Smith Thomas10, p. 113]) but a general Fréchet-Urysohn point may belong to $F_{q_*}$ . Being a point of countable tightness, however, it must belong to the closure of $L_{q_*}$ , and the set of such points must be countable (Corollary 4.6(i)).

The importance of Fréchet-Urysohn spaces is that every quotient of a compact first countable (or Fréchet-Urysohn) space is Fréchet-Urysohn (see Corollary 5.2). For example, let $Y=[0,1]\times [0,1]$ with the order topology of the lexicographic order and let $V=[0,1]\times \{0,1\}$ . Let $q:Y\to X$ be the quotient map that takes V to a point  $x_0$ . Then Y is compact, Hausdorff, and first countable, and V is closed but not a zero-set, so X is compact, Hausdorff and Fréchet-Urysohn but not first countable at  $x_0$ .

Recall that a topological space X is sequential if every sequentially closed subset of X is closed. Every Fréchet-Urysohn space is sequential, but there are compact Hausdorff sequential spaces which are not Fréchet-Urysohn. More generally, every sequential space has countable tightness; and the famous Moore-Mrowka problem—shown by Ostaszewski et al. to depend on set-theoretic axioms—was whether there exists a compact Hausdorff space of countable tightness which is not sequential.

If Y is a second countable locally compact Hausdorff then $\operatorname {Cl}(Y)$ is also second countable [Reference Fell9, Remark 3, p. 474], but the same does not hold for first countability. Indeed, if Y is a locally compact Hausdorff space, then the following three conditions are equivalent: (a) $\operatorname {Cl}(Y)\setminus \{\emptyset \}$ is first countable; (b) Y is both hereditarily Lindelof and hereditary separable (for a locally compact space, first countability is a consequence of being hereditarily Lindelof); and (c) $\operatorname {Cl}(Y)\setminus \{\emptyset \}$ is countably tight [Reference Hou12], [Reference Costatini, Hola and Vitolo8, Corollary 2.16]. The unit square with the lexicographic order (above) is neither separable nor hereditarily Lindelof. The passage from Y to $Y_*$ in Theorem 3.6 thus preserves second countability but it is not clear that it will preserve first countability or the Fréchet-Urysohn property (although we do not have specific counter-examples). This problem, however, is easily circumvented.

Arkhangel’skii’s question. In [Reference Arkhangel’skii5, Problems 5.14 and 5.15], Arkhangel’skii asked for a Hausdorff quotient of a locally compact second countable Hausdorff space which is (countable and) Fréchet-Urysohn but nowhere first countable. A locally compact second countable Hausdorff space is perfectly normal (i.e., every open set is an $F_{\sigma }$ ), and the same is true in every Hausdorff quotient, so every point in a quotient is a $G_{\delta }$ , and it is well known that a $G_{\delta }$ point with a compact neighborhood is first countable. Thus, the required example had to have $L_{q_*}$ empty and $F_{q_*}\cup N_{q_*}$ as a Fréchet-Urysohn space, along the lines of the space $S_{\omega }$ mentioned in Section 1.

In the absence of a structure theory for $k_{\omega }$ -spaces, Arkhangel’skii’s question is very pertinent, but in the course of this paper, we have seen multiple reasons why such an example cannot exist: for instance, every Fréchet-Urysohn point in a $k_{\omega }$ -space lies in the closure of $L_{q_*}$ (Corollary 4.6(ii)). As we have mentioned, the question had, in fact, already been answered negatively by Siwiec in 1976 [Reference Siwiec22] who showed (among other things) that if a $k_{\omega }$ -space is Fréchet-Urysohn and has a $k_{\omega }$ -decomposition of compact metric spaces then it is the closed image of a locally compact separable metric space (and hence Morita’s theorem implies that $L_{q_*}$ is dense).

Since a locally compact second countable space is hereditarily Lindelof, Corollary 5.3(i) and (ii) gives a more general negative answer to Arkhangel’skii’s question. If the Continuum Hypothesis is assumed, a still more general answer can be given.

In particular, under (CH), every Fréchet-Urysohn $k_{\omega }$ -space is first countable at a dense subset of points. On the other hand, in 1987 Malyhin used the method of forcing to exhibit a compact Hausdorff Fréchet-Urysohn which is nowhere first countable [Reference McCoy and Ntantu15]. Thus Corollary 5.3 may be about as far as one can go in ZFC in answer to Arkhangel’skii’s question.