I am wondering about the use of the word separablity in two different areas of mathematics, namely algebra and topology.
In topology, we call a topological space $(X,\mathscr{T})$ if it contains a dense subset, $\bar{D}=X$, with $d(X)$≤$\omega$ (with $|\mathbb{N}|=\omega$).
In algebra, I heard of separable polynomials $p \in K[X]$, $K$ being a field, meaning that the roots of those polynomials are all distinct in the algebraic closure $\bar{K}$, using the bar notation here too.
I cannot connect those two concepts. Therefore my question to you is:
Can you think of an example putting together the concept of polynomials, density and continouity and closure in order to connect those two different kinds of separablity or can you show me why it is not a good question to ask?
In particular I am wondering if one can use $\sigma$-finiteness (which in a way has some properties in common with separablity - the sets wont get too big, stay countable). So I'd be fine with something using $\sigma$-finiteness and measureability too (where separablity is used for Lusin's theorem as a condition that says that measureablity is at least close to continouity).