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3 votes
0 answers
26 views

Given a metric space $X$, is there a natural way to view the quasi-isometry group $QI(X)$ as a topological group?

Given a metric space $(X,d)$, we define $QI(X)$ as the set of quasi-isometries $f : X \to X$, modulo the equivalence relation $$ f \sim g \ \ \ \ \text{ if and only if } \ \ \ \sup_{x \in X} \ d(f(x)...
jpmacmanus's user avatar
0 votes
1 answer
98 views

Is every subgroup closed in this complete, nondiscrete topological group?

Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
Nick Belane's user avatar
2 votes
1 answer
49 views

Is any submetrizable linear topology linearly submetrizable?

Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$. Is ...
erz's user avatar
  • 5,529
0 votes
1 answer
311 views

Measure on group invariant under group action on metric space

This is a question very similar to I recently asked on mathexchange, but different enough to get its own entry in MO. The setting is still the same. I consider the metric space $\mathbb{R}$ and the ...
Sellerie's user avatar
  • 103
14 votes
2 answers
1k views

Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...
Jonathan Gleason's user avatar
10 votes
4 answers
2k views

When do isometric actions exist?

Let $X$ be a metrizable topological space and $G$ be a locally compact group. Given a continuous (left) action of $G$ on $X$, is there a metric on $X$, compatible with the topology, for which the ...
Kamran Reihani's user avatar