# Tagged Questions

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### Is there a subset of R such that their Cantor-Bendixson rank is the first limit ordinal?

I'm looking for a set $A \subset \mathbb{R}$ such that $\bigcap^\infty_{n=0} A^{(n)}$ is a perfect set (i.e $X'=X$) but $\forall n \in \mathbb{N}$ the set $A^{(n)}$ isn't perfect (where ...
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### Polish topological group

A friend asked me to help him prove that the topological group $\mathrm{Homeo}(0,1)$ (homeomorphism of $(0,1)$ with the compact open topology) is Polish (that is, separable and completely metrizable). ...
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### homeomorphism of cantor set extends to the plane?

Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ...
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### Prove that Baire space $\omega^\omega$ is completely metrizable?

When I tried to prove that Baire space $\omega^\omega$ is completely metrizable, I defined a metric $d$ on $\omega^\omega$ as: If $g,h \in \omega^\omega$ then let $d(g,h)=1/(n+1)$ where $n$ is the ...
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### Uncountably many non-homeomorphic compact subsets of the circle

As the title says, the question is whether there are uncountably many non-homeomorphic compact subsets of the unit circle. I'm assuming this is true, but I wouldn't mind an elegant proof.
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### Sets with the property of baire in proof of Poincare Recurrence Theorem

I have a problem with the proof of Thm 17.1 in Oxtoby's Category and Measure. In statement of the Poincare Recurrence Theorem he does not say anything about property of Baire, but in the proof he ...
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### Why is the boolean closure of $F_{\sigma}$-sets not in $F_{\sigma}\cap G_{\delta}$?

In the Borel hierarchy, why is the boolean closure of $F_{\sigma}$ or $G_{\delta}$ equal to $F_{\sigma \delta} \cap G_{\delta \sigma}$? If I take the complement of an element in $F_{\sigma}$ I got an ...
Let $A$ be subset of $\mathbb{R}$ that contains no nonempty perfect subsets. Is $A$ countable?