# Tagged Questions

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### Is the space $\mathbb N^ \mathbb N$ metrisable? [duplicate]

Given the space $\mathbb N^ \mathbb N$ with the topology generated by basis sets of the form: $$[V,n] = \{x \in \mathbb N^ \mathbb N ; V \text{ is an n prefix of x}\}$$ I can see that this space is ...
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### Dense subset of Cantor set homeomorphic to the Baire space

Does anyone know a proof that the Cantor set, $\{0,1\}^{\mathbb{N}}$, has a dense subset homeomorphic to the Baire space, $\mathbb{N}^{\mathbb{N}}$? Thank you.
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### Is it possible to have $D=\Bbb P$

Let $f:\Bbb R\to \Bbb R$ and $D=\{x\in \Bbb R: f$ is discontinuous at $x\}$. My problem is : Is it possible to have $D=\Bbb P$ where $\Bbb P$ is the set of irrationals in $\Bbb R$. I know the ...
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### Extending isometries between compact subspaces of Cantor space

Let $\omega$ be the set of natural numbers. $2^\omega$ is the Cantor space. Suppose $K$, $L \subset 2^\omega$ are compact, and there is an isometry $f: K \to L$. Then how could one extend $f$ to an ...
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### Continuous images of open sets are Borel?

Consider a Polish space $(X,d)$ and any metric space $(Y,e)$. If we have a continuous surjection $f:X\to Y$ then is the image $f(U)$ of any open subset $U\subset X$ a Borel set in $Y$? I know that ...
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### Extending a homeomorphism of a subset of a space to a $G_\delta$ set

I am having trouble figuring out the following question (3.10 in Kechris, Classical Descriptive Set Theory): If $X$ is completely metrizable, and $A\subseteq X$ with $f:A\to A$ a homeomorphism, then ...
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### Is it easier to make perfect sets using intersection [closed]

A perfect set $A$ is one in which every point is a limit point. So it has to be closed. Does this mean that if we want to generate perfect sets inductively it is usually best to just intersect ...
Let $X$ be complete separable metric space and $A\subset X$ is open. Does it mean that there is a compact subset of $A$? My solution is the following: since $A$ is open there is $B(x,r)\subset A$, ...