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.
EDIT: See also: https://math.stackexchange.com/a/1843091/327486.
These questions are also (indirectly) related: Is Every (Non-Trivial) Path Connected Space Uncountable?, Why study non-T1 topological spaces?, What is the motivation behind the arbitrary union topological axiom?