nLab topologically complete space

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 $C$, $M$, $S$, and $T$ be categories, and suppose that we have functors $U\colon C \to M$, $V\colon M \to S$, and $W\colon S \to T$. 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 $C$ as a subcategory of $M$, and we'll think of the objects of $C$ as the complete spaces in $M$.

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

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

An object $X$ of $S$ is $M$-izable if there is an object $Y$ of $M$ and an isomorphism in $S$ between $X$ and $V(Y)$. $X$ is completely $M$-izable if there is an object $Z$ of $C$ and an isomorphism in $S$ between $X$ and $V(U(Z)$. $X$ is $T$-ly $M$-izable if there is an object $Y$ of $M$ and an isomorphism in $T$ between $W(X)$ and $W(V(Y))$. Finally, $X$ is $T$-ly completely $M$-izable if there is an object $Z$ of $C$ and an isomorphism in $T$ between $W(X)$ and $W(V(U(Z)))$. (If all functors are isofibrations, then we may without loss of generality require these $T$-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 $M$ and $T$ rather than terms for $M$ and $T$ themselves; for example, in the standard example where $M$ is $Met$ and $T$ is $Top$, then we say that $X$ is topologically completely metrizable (to mean that $X$ is $Top$-ly completely $Met$-izable). $C, U, V, W$ are suppressed in the terminology because they are supposed to be obvious; $S$ is suppressed because you are supposed to know what it is when you first mention $X$.

Special cases

A space $X$ (an object of $S$) is $M$-izable if and only if $X$ is $S$-ly $M$-izable; $X$ is completely $M$-izable iff $X$ is $S$-ly completely $M$-izable. $X$ is complete iff $X$ is $S$-ly completely $S$-izable; $X$ is always $S$-ly $S$-izable. Finally, $X$ is $T$-ly completely $M$-izable iff $X$ is $T$-ly $C$-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, $M$ 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, $M$ 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, $M$ 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 $S$ must be specified explicitly; that is, we speak of the category of $T$-ly $M$-izable $S$-spaces or the category of $T$-ly completely $M$-izable $S$-spaces. However, if $S$ is not specified, then we take it to be $T$ by default, so the $T$-ly [completely] $M$-izable spaces are the same as the [completely] $M$-izable $T$-spaces. We consider these categories to be full subcategories of $S$.

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

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.

Related concepts

• metrisable space

• cotopology

• uniform space

References

• A.V. Arkhangel′skii (1977). Complete space. Matematicheskaya entsiklopediya. Updated English version.

• John Kelley (1955). General Topology, 1975.

• Eric Schechter (1997). Handbook of Analysis and its Foundations, Section 20.12.