The most natural way one can introduce topology, and what topology is about, is to explain what a metric space is. A metric space is a set, X, equipped with a metric d, which serves to measure distances between points.

The purpose of this chapter is to introduce the basic notions that we shall explore throughout this book, in this more familiar setting.

However, we must face a conundrum. Imagine we explained what an open subset is in a metric space. In the more general settings we shall explore in later chapters, we would have to redefine opens. Redefinitions are bad mathematical practice, and for a good reason: we would never know which definition we would mean later on; i.e., are the opens we shall use meant to be those introduced in the metric case here, or the more general kind introduced in later chapters? So we shall only talk about sequential opens, not opens, here. Sequential opens will provide particular examples of opens, which we hope should be illuminating. We proceed similarly for sequentially closed subsets. The adjective “sequential” is itself justified by the fact that these notions will be defined through convergence of sequences – and convergence is arguably one of the central concepts in topology.

Metric spaces

Let us describe what a metric space, i.e., a set X with a metric d, is, in intuitive terms first.

The following is a prototypical example of a metric space; whenever one looks for intuitions about metric spaces, we should probably first imagine what is happening there. This example is the real plane. Think of it as a blank sheet of paper, extending indefinitely left, right, up, and down. On this sheet of paper, one can draw points.

Topology, topological spaces

Abstracting away from metrics, a topological space is a set, with a collection of so-called open subsets U, satisfying the following properties. We have already seen them, for sequentially open subsets of a metric space, in Proposition 3.2.7.

Definition 4.1.1 (Topology) Let X be a set. A topology on X is a collection of subsets of X, called the opens of the topology, such that:

• every union of opens is open (including the empty union, Ø);

• every finite intersection of opens is open (including the empty intersection, taken as X itself).

A topological space is a pair (X,O), where O is a topology on X.

We often abuse the notation, and write X itself as the topological space, leaving O implicit. It is also customary to talk about the elements of a topological space X as points.

Example 4.1.2 The sequentially open subsets of a metric space form a topology. This is what Proposition 3.2.7 states exactly.

Given a metric space X, d, we shall call metric topology the topology whose opens are the sequentially open subsets.

Example 4.1.3 We shall often consider ℝ with the metric topology of its L1 metric. We shall either call this space ℝ, or ℝ with its metric topology when there is a risk of confusion as to which topology is intended. There are indeed other natural candidates, as we shall see in Example 4.2.19, for instance.

We recapitulate the axiomatic foundations on which this book is based in Section 2.1. We then recall a few points about finiteness and countability in Section 2.2 and some basics of order theory in Section 2.3, and discuss the Axiom of Choice and some of its consequences in Section 2.4. These points will be needed often in the rest of this book. If you prefer to read about topology right away, and feel confident enough, please proceed directly to Chapter 3.

Foundations

We shall rest on ordinary set theory. While the latter has been synonymous with Zermelo–Fraenkel (ZF) set theory with the Axiom of Choice (ZFC) for some time, we shall use von Neumann–Gödel–Bernays (VBG) set theory instead (Mendelson, 1997).

There is not much difference between these theories: VBG is a conservative extension of ZFC. That VBG is an extension means that any theorem of ZFC is also a theorem of VBG. That it is conservative means that any theorem of VBG that one can express in the language of ZFC is also provable in ZFC.

The main difference between VBG and ZFC is that the former allows one to talk about collections that are too big to be sets. This is required, in all rigor, in the definition of (big) graphs and categories of Section 4.12. VBG allows us to talk about, say, the collection V of all sets, although V cannot itself be a set. This is the essence of Russell's paradox: assume there were a set V of all sets.

In Section 4.2.2, we discussed how computer programs could be thought of as computing values x obtained as supn∈ℕxn, where xn are values in a given dcpo, e.g., the dcpo of sets of formulae in the example of the automated theoremproving computer. A distinguishing feature of the approximants xn to x is that they are finite, and this particular relation between xn and x can be described in arbitrary posets by saying that xn is way-below x. The way-below relation ≪ is a fundamental notion, leading to so-called continuous and algebraic dcpos, and we define it and study it in Section 5.1.

Beyond dcpos, the way-below relation will be instrumental in studying the lattice of open subsets of a topological space (Section 5.2). This will lead us to investigate the spaces whose lattice of open subsets is continuous, the so-called core-compact spaces. We shall see that the core-compact spaces are exactly the spaces X that are exponentiable, that is, such that we can define a topology on the space [X → Y] of continuous maps from X to an arbitrary space Y so that application and currification are themselves continuous (Section 5.3). These are basic requirements in giving semantics to higher-order programming languages, and desirable features in algebraic topology.

The way-below relation

The approximation, or way-below relation ≪ on a poset X is of fundamental importance in the study of dcpos.

Purpose This book is an introduction to some of the basic concepts of topology, especially of non-Hausdorff topology. I will of course explain what it means (Definition 4.1.12). The important point is that traditional topology textbooks assume the Hausdorff separation condition from the very start, and contain very little information on non-Hausdorff spaces. But the latter are important already in algebraic geometry, and crucial in fields such as domain theory.

Conversely, domain theory (Abramsky and Jung, 1994; Gierz et al., 2003), which arose from logic and computer science, started as an outgrowth of theories of order. Progress in this domain rapidly required a lot of material on (non-Hausdorff) topologies.

After about 40 years of domain theory, one is forced to recognize that topology and domain theory have been beneficial to each other. I've already mentioned what domain theory owes to topology. Conversely, in several respects, domain theory, in a broad sense, is topology done right.

This book is an introduction to both fields, dealt with as one. This seems to fill a gap in the literature, while bringing them forth in a refreshing perspective.

Secondary purpose This book is self-contained. My main interest, though, as an author, was to produce a unique reference for the kind of results in topology and domain theory that I needed in research I started in 2004, on semantic models of mixed non-deterministic and probabilistic choice.