Is TOP a small category?

Question

A quick question... is the category of topological spaces and continuous maps a small category? If so how do we know and if not how do we know not?

Answer 1

Since each set may be viewed as a topological space (e.g. via the discrete topology) there are "at least" as many topological spaces as sets; so no, Top is not small.

It is, however, locally small: for $X, Y$ fixed spaces, the collection $Hom(X, Y)$ of all continuous maps from $X$ to $Y$ is a set (note that there are at most $\vert Y\vert^{\vert X\vert}$-many of these).

Answer 2

Categories of sets with structure are basically never small — even something as simple as the collection of one-element sets is not small. (there are too many choices for what that single element can be!)

Some such categories can be essentially small, meaning that they are equivalent to a small category. For example, the category of finite dimensional vector spaces over $\mathbb{R}$ is not small, but it is essentially small.

However, Top is not even essentially small, for another simple set-theoretic reason: for every cardinal number, there is a discrete set of that cardinality, and these are all non-isomorphic. The collection of cardinal numbers is not small, and consequently the collection of objects of Top cannot be small.

Or put differently, Set itself is a full subcategory of Top, with one such embedding given by selecting the discrete topology (the indiscrete topology works too). Since Set isn't small, neither is Top.