When can one find a neighborhood for a point?
I figured out this confusion when considering the concept of locally Euclidean. $M$ is locally euclidean, if for every point of $M$ one can find a neighborhood, which is homeomorphic to an open ball in $\mathbb{R}^n$.
But when can one find such neighborhood? Or does the def. for locally Euclidean already presuppose that it exists (that is, it exists when one calls a set locally Euclidean, so the concept of locally Euclidean itself doesn't say about how/when to find such neighborhood).
$V$ is neighborhood of point $p$ if exists open ball with centre $p$ and radius $r$ s.t.
$$\{ x \in X : d(x,p) < r \}=B(p,r)$$
is contained in $V$.
Since I believe that one isn't constrained for e.g. $M= \emptyset$, then e.g. for this $M$ one cannot find a neighborhood?
Then what properties lead to such neighborhood existing? Connectivity etc.
There is no expectation of there being a recipe for finding such a locally Euclidean neighbourhood. Such a neighbourhood either exists for all points, or this is not the case. Mathematicians are mostly platonists. We can work with the assumption that such spaces exist and there are lots of motivating examples (like curves and surfaces) but for abstract spaces, if we call it locally Euclidean, then we just suppose we are given such neighbourhoods for each point, entirely in the abstract and then study the consequences that such a structure has on the properties of such a space etc.
If we want to apply such theory for concretely defined spaces we should indeed check that the definition holds for that space. This entails find neighbourhoods in the space and homeomorphisms from those neighbourhoods to Euclidean ones.