# Classification of points in a topological space

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:

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

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

3. 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.

• Something is wrong here. Taking U = V = X and $\phi$ to be the identity gives that every two points are equivalent. Mar 9 '17 at 1:16
• @Anonymous Presumably what is missing is $\phi(x) = y$. Mar 9 '17 at 1:24
• Hausdorff dimension is not a topological concept, as it is defined using a particular metric. Perhaps you mean topological dimension (of which there are several flavours). Mar 9 '17 at 1:33
• Thank you for the comments. Indeed, $\phi(x)=y$ is missing. I am correcting the question. Mar 9 '17 at 11:02
• A convergent sequence shows that [x] might not be open (or closed) even if $X$ is compact and metrizable. Mar 9 '17 at 15:30

1. Many invariants of algebraic topology have local versions. For example, local (co-) homology of $X$ at $x\in X$ is defined as the (co-) homology of the pair $(X,X\setminus\{x\})$, which is a local invariant by excision. If $X$ is an $n$-manifold, the only non-trivial local (co-) homology group is $H_n(X,X\setminus\{x\})\cong\mathbb Z$. On the other hand, boundary points in manifolds with boundary have trivial local homology. See for example Hatcher's book, p. 126. However, invariants like these work best for simplicial complexes.