I searched all over for an answer to this question back in my student days. I found the answer in a paper by Sierpinski, "Sur une série potentielle qui, étant convergente en tout point de son cercle ...

For another proof that your question has a positive answer, you may look at the folowing paper by Jean Saint Raymond: http://www.math.jussieu.fr/~raymond/preprints/inversion.dvi It seems that he was ...

Even with a homeomorphism, preserving $=^*$ does not imply reflection. There might be an easy example, I'm not sure; at any rate it follows from a result in topological dynamics (the "absorption ...

A way to prove it: 1/ any set of the form $A_1 \times \mathbb R \ldots \times \mathbb R$, where $A_1$ is Borel, or more generally a "Borel rectangle" with only one slice not equal to the whole space, ...

Another rather nice example: the group of automorphisms of $[0,1]$ endowed with the Lebesgue measure (i.e, bi-measurable maps preserving the measure, identified when they coincide outside a set of ...

The hyperspace of any Peano continuum (locally connected metric continuum) is homeomorphic to the Hilbert cube; this is a result of Curtis and Schori, see here. I learnt about this result in a paper ...

The meet of the left- and right- uniformities is called the Roelcke uniformity, as Todd Eisworth mentions. The topology it generates is the original topology (the same is true for the join of the two ...

It is not so easy to construct an example of this kind, I think, because of the Hurewicz theorem: if $X$ is a coanalytic separable metrizable separable space, then either it is Polish (in particular, ...

I got interested in a similar problem (not quite the same thing, though) a few years back but could not find anything really interesting. Here is what I found in the litterature; In case $X$ is ...

This space is certainly not compact in general, unless I misunderstood your question. For instance, assume the probability space in question is $[0,1]$ with Lebesgue measure. Then endow the isometry ...