Order, Distance, Closure and Convergence: Reconciling Competing Fundamental Topological Concepts

Already in Hausdorff’s 1914 book, often considered the cradle of general topology, one finds traces of a discussion on the relative strengths of the concepts mentioned in the title of this talk. In fact, one may argue that Hausdorff anticipated the basic ideas of how to unify these concepts, which were developed only later on by many mathematicians over the course of a century, as propagated in Hofmann, Seal & Tholen. Indeed, Hausdorff thought of ordering points by assigning to every pair of them a (truth) value, just as a metric assigns to them a number. More importantly, he also contemplated extending such assignments to pairs, whose second component would remain a point, but whose first component would now be a subset, or a sequence, of points of the space in question, which is then assigned a value that measures the extent to which that point lies in the closure of the subset, or is a convergence point of the sequence. In monoidal topology, the first components of the arguments of such value assignments are given by a monad T on Set (where, for a set X, TX could be all strings of points of X, or all subsets of X, or all filters on X, etc), while the values themselves must lie in a quantale V (which could be the lattice 2 = {true, false}, or the non-negative extended real line, or the lattice of distribution functions of that extended line, etc). These structures must then satisfy two basic axioms, generalizing the reflexivity and transitivity of relations. With morphisms to be maps laxly preserving the structure, this defines the category (T, V )-Cat, which is topological over Set and, therefore, automatically boosts a wealth of good properties. The principal categories of interest to topologists are all of this type, or may be reflectively or coreflectively embedded into them. But as indicated above, an individual category, like Top, may be presentable in various (T, V )-guises, and establishing the equivalence may not necessarily be easy. In fact, its validity may depend on additional properties of V . For example, for T the powerset monad, we may easily extend the usual properties of distance and closure to define and study so-called V -topological spaces, but the establishment of their equivalent description in terms of a V -valued ultrafilter convergence relation requires V to be completely distributive (see Lai and Tholen). Among other theorems, we will present this equivalence statement and show how it unifies previous results for topological spaces and approach spaces and leads to novel applications. Time permitting we will also discuss essential topological properties, like compactness and separation, in the V -context

Enriched Topology and Asymmetry

Mathematically modeling the question of how to satisfactorily compare, in many-valued ways, both bitstrings and the predicates which they might satisfy-a surprisingly intricate question when the conjunction of predicates need not be commutative-applies notions of enriched categories and enriched functors. Particularly relevant is the notion of a set enriched by a po-groupoid, which turns out to be a many-valued preordered set, along with enriched functors extended as to be variable-basis . This positions us to model the above question by constructing the notion of topological systems enriched by many-valued preorders, systems whose associated extent spaces motivate the notion of topological spaces enriched by many-valued preorders, spaces which are non-commutative when the underlying lattice-theoretic base is equipped with a non-commutative (semi-)tensor product. Of special interest are crisp and many-valued specialization preorders generated by many-valued topological spaces, orders having these consequences for many-valued spaces: they characterize the well-established L-T0 separation axiom, define the L-T1(1) separation axiom-logically equivalent under appropriate lattice-theoretic conditions to the L-T1 axiom of T. Kubiak, and define an apparently new L-T1(2) separation axiom. Along with the consequences of such ideas for many-valued spectra, these orders show that asymmetry has a home in many-valued topology comparable in at least some respects to its home in traditional topology

Topology from enrichment: the curious case of partial metrics

For any small quantaloid \Q, there is a new quantaloid \D(\Q) of diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa), and then it is particularly interesting to compare categories enriched in \Q with categories enriched in \D(\Q). Taking Lawvere's quantale of extended positive real numbers as base quantale, \Q-categories are generalised metric spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a revised section on Cauchy sequences in a partial metric spac

A categorical approach to the maximum theorem

Berge's maximum theorem gives conditions ensuring the continuity of an optimised function as a parameter changes. In this paper we state and prove the maximum theorem in terms of the theory of monoidal topology and the theory of double categories. This approach allows us to generalise (the main assertion of) the maximum theorem, which is classically stated for topological spaces, to pseudotopological spaces and pretopological spaces, as well as to closure spaces, approach spaces and probabilistic approach spaces, amongst others. As a part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for publication in JPA

A Note on the Topologicity of Quantale-Valued Topological Spaces

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr's characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for ${\sf{V}}$ completely distributive, ${\sf{V}}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to ${\sf{V}}$-valued relations. As a consequence, ${\sf{V}}$-$\bf Top$ is seen to be a topological category over $\bf Set$, provided that ${\sf{V}}$ is completely distributive. In this paper we give a choice-free proof that ${\sf{V}}$-$\bf Top$ is a topological category over $\bf Set$ under the considerably milder provision that ${\sf{V}}$ be a spatial coframe. When ${\sf{V}}$ is a continuous lattice, that provision yields complete distributivity of ${\sf{V}}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted

A Note on the Topologicity of Quantale-Valued Topological Spaces

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of${\sf{V}}$-valued topological spaces may be introduced as a full subcategory ofthose ${\sf{V}}$-valued closure spaces whose closure operation preserves finitejoins. In generalization of Barr's characterization of topological spaces asthe lax algebras of a lax extension of the ultrafilter monad from maps torelations of sets, for ${\sf{V}}$ completely distributive,${\sf{V}}$-topological spaces have recently been shown to be characterizable bya lax extension of the ultrafilter monad to ${\sf{V}}$-valued relations. As aconsequence, ${\sf{V}}$-$\bf Top$ is seen to be a topological category over$\bf Set$, provided that ${\sf{V}}$ is completely distributive. In this paperwe give a choice-free proof that ${\sf{V}}$-$\bf Top$ is a topological categoryover $\bf Set$ under the considerably milder provision that ${\sf{V}}$ be aspatial coframe. When ${\sf{V}}$ is a continuous lattice, that provision yieldscomplete distributivity of ${\sf{V}}$ in the constructive sense, hence also inthe ordinary sense whenever the Axiom of Choice is granted