Let $X$ be a topological space, which can be assumed locally compact, second countable or Hausdorff, if necessary. For a pair of points $x,y\in X$ we write $x\sim y$ if they are related by a local homeomorphism, i.e., if there exist open neighbourhoods $U\ni x$ and $V\ni y$, and a (local) homeomorphism $\phi:U\to V$ such that $\phi(x)=y$. This is an equivalence relation, and I am interested in the quotient set $X/[\sim]$, i.e., the classification of all points of $X$ into equivalence classes.

**Questions:**

Are there good invariants to parameterize $X/[\sim]$? Maybe some topological dimensions, or something else?

Are equivalence classes $[x]$ open, closed or somehow nice?

Let $\operatorname{Hom}(X)$ be the group of global homeomorphisms of $X$. When is the action of $\operatorname{Hom}(X)$ in each(some) equivalence class(es) $[x]$ transitive? In other words, are there good criteria for local homeomorphisms to extend to global ones?

I am interested in minimal assumptions on $X$ that make the answer to each question affirmative, rather than counterexamples if they fail in this generality. Instructive examples are also welcome, though.

Please, note that I am no expert in topology, so I prefer more accessible language and references. Thank you.