nLab
topologically complete space

Topologically complete spaces

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Topologically complete spaces

Idea

A topological space is topologically complete if and only if it is completely metrizable. However, the term ‘topologically complete’ may be applied to other spaces as well, and in this case it means that the underlying topological space is completely metrizable.

One can vary this by considering some other kind of underlying space (besides the underlying topological one) and other kinds of completability (besides complete metrizability).

Definitions

General

Let CC, MM, SS, and TT be categories, and suppose that we have functors U:CMU\colon C \to M, V:MSV\colon M \to S, and W:STW\colon S \to T. We will think of the objects of these categories as spaces, and we'll think of these functors and their composites as forgetful functors. Also, we'll think of CC as a subcategory of MM, and we'll think of the objects of CC as the complete spaces in MM.

The basic example to keep in mind is the case where TT is the category Top of topological spaces and continuous functions, MM is the category Met of metric spaces and short functions, and CC is the full subcategory of MetMet consisting of the complete metric spaces. In the motivating example, SS is also MetMet, but this can remain flexible. In any case, U,V,WU, V, W should all be standard forgetful functors.

Even in other situations, it should be that once you specify M,S,TM, S, T, then C,U,V,WC, U, V, W should all be obvious. Typically, C,M,S,TC, M, S, T will be concrete categories, U,V,WU, V, W will be faithful isofibrations, and UU will additionally be full, but this doesn't really affect anything.

An object XX of SS is MM-izable if there is an object YY of MM and an isomorphism in SS between XX and V(Y)V(Y). XX is completely MM-izable if there is an object ZZ of CC and an isomorphism in SS between XX and V(U(Z)V(U(Z). XX is TT-ly MM-izable if there is an object YY of MM and an isomorphism in TT between W(X)W(X) and W(V(Y))W(V(Y)). Finally, XX is TT-ly completely MM-izable if there is an object ZZ of CC and an isomorphism in TT between W(X)W(X) and W(V(U(Z)))W(V(U(Z))). (If all functors are isofibrations, then we may without loss of generality require these TT-isomorphisms to be equalities, which is typically what is done in topology books.)

The actual terminology used will be based on terms for the objects of MM and TT rather than terms for MM and TT themselves; for example, in the standard example where MM is MetMet and TT is TopTop, then we say that XX is topologically completely metrizable (to mean that XX is TopTop-ly completely MetMet-izable). C,U,V,WC, U, V, W are suppressed in the terminology because they are supposed to be obvious; SS is suppressed because you are supposed to know what it is when you first mention XX.

Special cases

A space XX (an object of SS) is MM-izable if and only if XX is SS-ly MM-izable; XX is completely MM-izable iff XX is SS-ly completely MM-izable. XX is complete iff XX is SS-ly completely SS-izable; XX is always SS-ly SS-izable. Finally, XX is TT-ly completely MM-izable iff XX is TT-ly CC-izable.

A space is topologically complete if it is topologically completely metrizable; this is the standard example above. At least one reference (Schechter 1997) uses this term for a space that is topologically completely pseudometrizable (that is, MM is the category of pseudometric spaces). According to Wikipedia, at least one reference (Kelley 1955) uses this term for a space that is topologically completely uniformizable (that is, MM is the category of uniform spaces).

A space is Dieudonné-complete if it is topologically completely uniformizable. At least one reference (Arkhangel′skii 1977) uses this term for a space that is topologically completely Hausdorff-uniformizable (that is, MM is the category of of Hausdorff uniform spaces).

Categories of complete spaces

If we consider the entire category of complete spaces in some sense, then SS must be specified explicitly; that is, we speak of the category of TT-ly MM-izable SS-spaces or the category of TT-ly completely MM-izable SS-spaces. However, if SS is not specified, then we take it to be TT by default, so the TT-ly [completely] MM-izable spaces are the same as the [completely] MM-izable TT-spaces. We consider these categories to be full subcategories of SS.

Up to equivalence of categories, the category of MM-izable TT-spaces (aka the TT-ly MM-izable spaces, that is the TT-ly MM-izable TT-spaces) is the category whose objects are taken from MM and whose morphisms are taken from TT; similarly, the category of completely MM-izable TT-spaces has objects taken from CC and morphisms taken from TT.

In particular, following our standard example, the category of topologically complete spaces (that is the category of completely metrizable toplogical spaces) may be taken to have complete metric spaces as objects and continuous functions as morphisms.

References

Last revised on June 25, 2017 at 09:56:13. See the history of this page for a list of all contributions to it.