TITLE: Is TOP a small category?
QUESTION [7 upvotes]: 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?

REPLY [15 votes]: 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.