TITLE: Is the set of all topological spaces bigger than the set of all metric space?
QUESTION [23 upvotes]: I was wondering right that since the notion of a topology is much more general than that of a metric, and that "neighborhodness", if you will, and the concept of continuity, is generalized by the notion of a topology. so is the set of all topological spaces actually bigger than that of metric spaces? in other words if $$\mathscr{T}=\{x|x \text{ is a topological space}\}$$
and if $$\mathscr{M}=\{x|x\text{ is a metric space}\}$$ then is $$\text{card}\mathscr{T}>\text{card}\mathscr{M}$$ or are they equal? in both cases how can we prove that?
or perhaps the sets $\mathscr{T}$ and $\mathscr{M}$ don't even exist at all similar to how the set of all sets doesn't exist?

REPLY [3 votes]: The other answers here deal (correctly) with cardinality questions about the classes you're asking about. I don't think that's the way to ask the question that seems to interest you. Perhaps you want to know in what sense there are more topological spaces than metric spaces. If that's your question then ...
Every metric space is a topological space in a natural way - a metric determines a topology. (Different metrics on the same space may or may not determine the same topology.) But there are topological spaces where the topology does not come from any metric. See https://en.wikipedia.org/wiki/Metrization_theorem and https://mathoverflow.net/questions/52032/examples-of-non-metrizable-spaces for examples.