Is there an analogous notion of cardinality for homeomorphisms (or other structure-preserving bijections)?

Question

As far as I know, we can assign an arbitrary set a cardinality, which is defined in such a way that if there is an injective function $A \to B$ then $A$ has smaller or equal cardinality than $B$, and if there is a bijection, then their cardinality is equal.

Question: Is there a similar notion of "topological cardinality" in topology, where we restrict these injective/bijective functions to be continuous?

Bonus question: Is there a similar notion of "$\mathcal C$-cardinality" in all categories $\mathcal C$?

It depends on what you mean by "similar." Obviously, the existence of a bijection means that the two structures must have the same cardinality. But if we're just talking about (simple) cardinality, any two sets are comparable -- either one is larger than the other (meaning there's an injection of the smaller into the larger) or both have the same size. In pretty much any other category I can think of, there will be incomparable structures. — Robert Shore, Aug 22 '24 at 15:37
Are you familiar with topological notions of dimension? But keep in mind that a continuous surjection can increase the dimension. — Moishe Kohan, Aug 22 '24 at 15:39
See the references in my answer to Can two topological spaces surject onto each other but not be homeomorphic?. — Dave L. Renfro, Aug 22 '24 at 15:43

score 2 · Answer 1 · answered Aug 23 '24 at 00:00

The topological equivalent to a bijection of sets is a homeomorphism, which is defined as a continuous bijection with continuous inverse. The existence of a homeomorphism between two topological spaces allows you to think of them as "the same space" in almost every respect.

I think the closest you'll get to a topological equivalent of an injective function of sets an embedding: A continuous function that is a homeomorphism with its image. Having an embedding allows you to think of one topological space as "living inside" another (sometimes called a "subspace").

Note that the "continuous inverse" and "homeomorphism with its image" parts of the definitions above are both important: An uncountable set with the discrete topology has a continuous bijection into $\mathbb{R}$, but we don't want to think of the two as the same space, nor do we want to think of an uncountable discrete space as a subspace of $\mathbb{R}$.

Unfortunately, these notions do not give you a nice equivalent of set cardinality, which you can think of as an intuitive "ordering" or "sizing" of the category. For example, an open interval and a closed interval are not homeomorphic, but each of them has an embedding into the other - so you can't really say one is "bigger" in a meaningful way.

I don't believe there's any way to get a "sizing" of topological spaces that's anywhere close to being as nice as cardinality.

Is there an analogous notion of cardinality for homeomorphisms (or other structure-preserving bijections)?

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