What's all the fuss about local bases? I find that everywhere I look people are confused of the notion of a local base, and frankly I am as well, because it seems to me it's equivalent an incredibly simple formulation, but everyone else expounds endlessly on "filters" and "filter bases" and comes up with these elaborate logical statements.
The definition of a "local base" that I've learned is that if $x\in X$ where $(X,\ \tau)$ is a topological space, and $N_x$ is the neighbourhood filter of $x$ (that is, the set of neighbourhoods of $x$), then a local base of $x$ is a subset $B_x\subset N_x$ such that $\forall A\in N_x$ there is some $B\in B_x$ such that $B\subseteq A$.
Alright, so, why don't we take $B_x$ to be the set of open sets containing $x$? That makes perfect sense to me, since every neighbourhood must contain some open set, and so the above logical formulation of a local base is satisfied. There you go, boom. Nothing special. No filters or anything. Clean and simple.
So why do topologists make such a fuss of local bases? There's all this stuff on "first countable" and "second countable" and it seems all the separation axioms of topological spaces are rooted in the idea of bases (some local, some global, from what I can tell). The definition of a locally convex TVS uses local bases. I see tons of people confused over what a local base is.
What makes local bases important? Why are people so confused, myself included over them? Why is such a complicated definition necessary? Is it because the $B_x$ I described isn't the only local base?
Best Answer
Frequently, there are too many open sets. It is nice to be able to verify topological assertions (like continuity, for example) in terms of a smaller subclass which is more manageable.
As an analogy, consider the notion of basis of a vector space. It is quite good that we can, for instance, have a linear map entirely determined by its values on a basis. The concept of basis helps us to represent the whole structure in terms of a smaller, hopefully more manageable subclass. This is the idea.