The simplest of the continuity theorems considered states that a Baire-measurable function between metric spaces has only a meagre set of discontinuity points. Results on Baire continuity (again, this theme goes back to Banach’s book) are given, for instance the Baire homomorphism theorem states that a Baire homomorphism between normed groups X, Y with X topologically complete is continuous. Another generalization is presented as Banach’s continuous-homomorphism theorem. The coincidence theorems we present derive from Sandro Levi’s 1983 result on the comparison of topologies, to the effect that if one refines the other, they must coincide on a subspace.

This starred (omittable) chapter is devoted to non-separable versions of results already proved in the more tractable separable context. As the star indicates, the results here are aimed more at the specialist topologist than at the general mathematical reader, our usual intended audience.

As well as BGT, the other main influence on this book is Oxtoby’s Measure and Category: A Survey of the Analogies between Topological and Measures Spaces (Springer, 1971). For Oxtoby, (Lebesgue) measure is primary, (Baire) category is secondary. Our view, as our title shows, reverses this. The book may thus be regarded as an extended demonstration of the power and wide applicability of the Baire category theorem. Chapter 2 – where we use ‘meagre’ and ‘non-meagre’ for ‘of first (Baire) category’ and ‘of second category’ – proves and discusses several versions of Baire’s (category) theorem: on the line, the intersection of any sequence of dense open sets is dense. We also discuss Baire measurability, and the Baire property. We likewise give a full treatment of the Banach category theorem – a union of any family of meagre open sets is meagre – also used extensively in the book. We discuss countability conditions, and games of Banach–Mazur type. The chapter ends with a discussion of p-spaces (plumed spaces).

This chapter may be viewed as a brief treatment of such parts of descriptive set theory as are needed in the main body of the text. The Borel hierarchy and analytic sets (Chapter 1) are developed further. The theorems of Souslin (analytic plus co-analytic imply Borel), Nikodym (preservation of the Baire property under the Souslin operation) and Marczewski (preservation of measurability under the Souslin operation) are stated (proved in more generality in Chapter 12). The Cantor Intersection Theorem is extended from closed (or compact) sets to analytic sets (Analytic Cantor Theorem). The Borel hierarchy is extended to the projective hierarchy: starting with the analytic sets $\sum^1_1$, their complements $\prod^1_1$ and the intersection of these, $\Delta^1_1$ (the Borel sets), one proceeds inductively: $\sum^1_{n+1}$ contains projections of $\prod^1_n$; their complements give $\prod^1_{n+1}$; intersections of these give $\Delta^1_{n+1}$, etc. The special importance of $\Delta^1_2$ is discussed.

The importance of infinite combinatorics is indicated by the book’s subtitle. Category (and indeed measure) methods are particularly useful for establishing generic behaviour: showing that a particular property predominates, without needing to (or indeed, being able to) show any specific example. Results of this type proved here include the Generic Dichotomy Principle, Generic Completeness Principle, Kestelman–Borwein–Ditor Shift-Compactness Theorem (used many times and abbreviated to KBD) and Kemperman’s Displacement Theorem.

The KBD theorem is about embedding subsequences of shifts of a suitably regular set into some target set. Developing work of Kingman (1963, 1964), we extend this here to embedding into all members of a family of sets. Useful here is the idea of shift-compactness. We also begin to pass effortlessly between the category and measure cases by working bitopologically, using the Euclidean topology for the category case and the density topology (Chapter 7) for the measure case.

Group-norms are vector-space norms but with the scalars restricted to units (invertibles), ±1. The Birkhoff–Kakutani theorem (a first-countable Hausdorff topological group has a right-invariant metric) we view as a normability theorem rather than a metrization theorem, a relative of Kolmogorov’s normability theorem for topological vector spaces (the condition for whose normability is that the origin have a convex bounded neighbourhood). The groups here need not be abelian, so one has left-sided and right-sided versions. Proved here is the Analytic Baire Theorem: if a normed group contains an (either-sided) non-meagre analytic set, it is Baire, separable and (modulo a meagre set) itself analytic. Other results here include the ‘Analytic Shift Theorem’ and the ‘Squared Pettis Theorem’, category relatives of the Steinhaus Difference Theorem.

The infinite combinatorics developed in the previous chapters may be harnessed to give a treatment of regular variation in quite general contexts. Particularly useful tools here are the Category Embedding Theorem and the Effros Theorem. The main theorems of regular variation (see, e.g., BGT) include the Uniform Convergence Theorem (UCT) and the Characterization Theorem. The UCT is extended to the $L_1$-algebra of a locally compact metric group, using Reiter-like conditions from amenability. The Characterization Theorem can be formulated for normed groups X and H, with T a connected non-meagre Baire subgroup of the group of homeomorphisms from X to H. If for h : X → H is Baire and $h(tx)h(x)^{-1} \rightarrow k(t)$ for x → ∞ in X, then k is a continuous homomorphism from T to H. A calculus of regular variation is developed, involving the ‘differential modulus’. The theory is extended to the case of non-commutative H.

The Category Embedding Theorem (CET) is a result in infinite combinatorics related to the Kestelman–Borwein–Ditor Theorem KBD, and also to the concept of shift-compactness. The relationships between KBD, CET and various forms of No Trumps NT are given.

The text proper of the book begins with Littlewood’s three principles. The first – ‘any measurable set is nearly a finite union of intervals’ – is essentially regularity of Lebesgue measure. The second – ‘any measurable function is nearly continuous’ – is Lusin’s Theorem. The third – ‘any convergent sequence of measurable functions is nearly uniformly continuous’ – is Egorov’s Theorem. Then what will be needed from general topology is summarised, with references, going as far as para-compactness. Modes of convergence – in measure (in probability), almost everywhere (almost sure), etc. – are discussed. The Borel hierarchy – the result of applying, to (say) the open sets, the sigma and delta operations (union and intersection) alternately – is developed, as far as the Souslin operation. Analytic sets – much used in the book – are briefly treated here.

As a counterpart to Chapter 9 on category–measure duality, we focus here on a variety of situations in which duality fails. We cover a range of topics: Liouville numbers, Banach–Tarski (‘paradoxical’) decompositions, restriction and continuity, random series, normal numbers, topological and Hausdorff dimension, random Dirichlet series, filters, genericity, the Fubini and Kuratowski–Ulam theorems. We give an account of modern results on forcing, deferring technicalities to Chapter 16.

The other prime example of a fine topology is the fine topology of potential theory (in the usual sense of electromagnetism, gravitation, etc.) This is finer than the Euclidean topology but coarser than the density topology. Each of these three topologies has its σ-ideal of small sets: the meagre sets for the Euclidean case, the polar sets for the fine topology of potential theory, and the (Lebesgue-)null sets for the density topology. The polar sets have been extensively studied, not only in potential theory as above but in probabilistic potential theory; pioneers here include P.-A. Meyer and J. L. Doob. Relevant here are the links between martingales and harmonic functions (likewise their sub- and super-versions), Green functions, Green domains, Markov processes, Brownian motion, Dirichlet forms, energy and capacity. The general theory of such fine topologies involves such things as analytically heavy topologies, base operators, density operators and lifting.

Steinhaus’ Theorem of Chapter 9, an interior-point result, was extended from the line under Lebesgue measure to topological groups under Haar measure by Weil. The resulting Steinhaus–Weil theory, which is extensive, is presented in Chapter 15. The Simmons–Mospan converse gives the condition for the extension to hold: in a locally compact Polish group, a Borel measure has the ‘Steinhaus–Weil property’ if and only if it is absolutely continuous with respect to Haar measure. We define measure subcontinuity (adapted from Fuller’s subcontinuity for functions), and amenability at the identity. We prove Solecki’s interior-point theorem: in a Polish group, if a set E is not left Haar null, then the identity is an interior point of $E^{-1}E$. Related results on sets such as ${AB}^{-1}$ rather than ${AA}^{-1}$ are given.

A point is a density point of a set if the ratio of the length of its intersection with an interval containing it to that of the interval tends to 1 as the interval shrinks to the point. The classical Lebesgue Density Theorem states that almost all points of a measurable set are density points. Declaring a set open when all its points are density points leads to a topology, the density topology. This is a fine topology – it refines the ordinary (Euclidean) topology, in having more open sets. The density-meagre sets are the Lebesgue-null sets. This result shows how working bitopologically – switching between the Euclidean and density topologies – enables us to switch between the category and measure cases. A list of properties of the line under the density topology is given. Caution is needed: for instance, the line is a topological group under the Euclidean topology, but not (only a paratopological group) under the density topology (as now multiplication is only separately but not jointly continuous).

We begin with the canonical status of the reals: this extends up to uniqueness to within isomorphism as a complete Archimedean ordered field, but not up to cardinality aspects. We discuss four ‘elephants in the room’ here (an elephant in the room is something obviously there but which no one wants to mention). The first elephant (from Gödel’s incompleteness theorem and the Continuum Hypothesis, CH): one cannot properly speak of the real line, but rather which real line one chooses to work with. The second is ‘which sets of reals can one use?’ (it depends on what axioms of set theory one assumes – in particular, the role of the Axiom of Choice, AC). The third is that there are sentences that are neither provable nor disprovable, and that no non-trivial axiom system is capable of proving its own consistency. Thus, we do not – cannot – know that mathematics itself is consistent. The fourth elephant is that even to define cardinals, the concept of cardinality needs AC.