"Set" vs "Space" in definitions?

Question

I have seen many definitions in two versions: sometimes referred to sets, sometimes to spaces.

Some examples: closed set/space, compact set/space, $F_{\sigma\delta}$ set/space.

I asked one of my lecturers about this and they said that if I introduce some definition with "space", I need to explain it more. So I am thinking, defining something as a set is more general, while defining on a space requires looking also at the space structure? Or does this depend on particular case?

I am particularly interested in the case with $F_{σδ}$ sets vs spaces, but I think it is important question in general.

Thank you for your advice.

Answer 1

The difference is simply that "space" is a notion that implies some underlying structure to it. A set may not have that structure.

This is particularly key when we begin to talk of subsets versus subspaces. Take $\Bbb R^n$, a vector space, as an example. There are a lot of subsets there: pick any number of elements you want at random, but odds are, they won't satisfy the vector space axioms and thus be a (vector) subspace.

What this "structure" is depends on the context (vector spaces, topological spaces, measure spaces, etc.), however. But regardless, when saying "space", you're implying the existence of some kind of underlying structure -- your instructor is poking at that fact: what is that structure?

It's a bit confusing since, especially colloquially, the terms often are used in about the same way. However, there is that technical and important difference there.

Answer 2

The terminology "a closed set" etc, is sloppy, but, alas, common. The better one is "a closed subset" since it makes sense only relative to an ambient topological space. Similarly, use "compact subset" or "compact space." Ditto $F_\sigma$ and $G_\delta$ subsets. In general, when a topologist says "a space" (without any further adjectives), they mean "a topological space." This is very common, get used to it. There are also $G_\delta$-spaces. This notion is different from the one of $G_\delta$-subsets in a topological space.