# Are there any “default” properties which hold for almost all topological spaces in analysis?

What is the simplest/most general commonly used (type of) topological space in analysis?

For instance, every example I can think of in analysis is first-countable.

I don't think it can be metric spaces, since there are non-metrizable spaces (for example spaces of weak-* functionals) of use in analysis.

I was thinking perhaps separable topological vector spaces which are either T0, T1, or T2. (I'm not sure which separation axiom is most appropriate though.)

Finally, which connectedness/compactness properties would we want?

Would local path-connectedness (or even local connectedness) be too strong in general? What about local compactness or paracompactness? Local finiteness?

EDIT: Considering the points brought up by Calvin Khor and Qiyu Wen below, I changed "separable" to "first-countable".

Related, but different question: Measuring the set-theoretical complexity of sets/spaces encountered in general analysis
In contrast, I am asking about topological properties of spaces in analysis.

• $L^∞$ is not separable. – Calvin Khor Jun 14 '16 at 23:31
• For a topological group $T_0$ implies $T_2$(Hausdorff) and $T_{3½}$ (completely regular). Basically because the group structure makes the space homogeneous. In particular this holds for all topological vector spaces. – Jyrki Lahtonen Jun 15 '16 at 5:04