Countable Sets in utility theory

Question

What does it mean for a quotient space to be "at most countable"? I read that in a Theorem and I didn't understand the terminology. Thank you in advance!

Answer 1

When we say that a space is at most countable, what we mean is that it has at most countably many (= finitely many or countably many) points. For example, $\mathbb{N}$ with the discrete topology is an at most countable space, since the set of points is $\mathbb{N}$ and $\mathbb{N}$ is countable. But the usual topology on $\mathbb{R}$ is not at most countable, since the set of reals is uncountable.

It's important to note that "quotients get smaller:" an uncountable space can have an at-most-countable quotient. The most obvious way for this to happen is if we collapse the whole space to a single point.