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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.

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    $\begingroup$ Something is wrong here. Taking U = V = X and $\phi$ to be the identity gives that every two points are equivalent. $\endgroup$ – Anonymous Mar 9 '17 at 1:16
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    $\begingroup$ @Anonymous Presumably what is missing is $\phi(x) = y$. $\endgroup$ – Robert Israel Mar 9 '17 at 1:24
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    $\begingroup$ 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). $\endgroup$ – Robert Israel Mar 9 '17 at 1:33
  • $\begingroup$ Thank you for the comments. Indeed, $\phi(x)=y$ is missing. I am correcting the question. $\endgroup$ – Bedovlat Mar 9 '17 at 11:02
  • $\begingroup$ A convergent sequence shows that [x] might not be open (or closed) even if $X$ is compact and metrizable. $\endgroup$ – Ramiro de la Vega Mar 9 '17 at 15:30
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Given the generality of the question, I can only give a partial answer.

  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.

  2. Exercise 28 on p. 133 gives equivalence classes that are locally closed. But again, the situation there is rather special.

  3. Even for manifolds, there is no general statement. Consider for example the disjoint union of a sphere and a torus.

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  • $\begingroup$ Thank you for the answer. Regarding Question 3., I agree, in the present formulation it may bee too hard to answer. In fact, transitivity in every connected component is good as well. Even a decomposition of every connected component into a finite number of orbits is not bad for me. I am not editing my question regarding this in order not to invalidate your answer. This answer is useful, but clearly not final. $\endgroup$ – Bedovlat Mar 9 '17 at 20:52

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