5
votes
0answers
70 views

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 ...
3
votes
2answers
165 views

Confusion in proof that $C(X,Y)$ is separable

From Kechris' Classical Descriptive Set Theory: (4.19) Theorem If $X$ is compact metrizable and $Y$ is Polish, then $C(X,Y)$ is Polish. In the proof of separability, we consider $$ C_{m,n} = \{f ...
4
votes
3answers
76 views

Open neighborhoods of a $G_\delta$ set

This may have a simple answer, but I couldn't find it so far either in textbooks or in math.stackexchange. Let $X$ be a metric space, and $$A=\bigcap^\infty_{n=1}A_n$$ a $G_\delta$ subset of $X$, ...
3
votes
1answer
77 views

Is a Borel subset in an analytic subset of a Polish space still analytic?

I encounter some problem like this Assume $A$ is a Borel subset of $B$. $B$ is an analytic subset of a Polish space $C$. Is $A$ an analytic set in $C$? while reading a book. But I don't know the ...
5
votes
2answers
210 views

Is any compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?

Every compact metric totally disconnected perfect space is homeomorphic to a Cantor space. Is every compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space? ...
4
votes
1answer
514 views

How to show that Hausdorff distance is a metric on the set of all compact non-empty subsets of a Polish space?

For each perfect Polish space $X$, let $H[X]$ be the set of all compact non-empty subsets of $X$. If $x ∈ X$ and $A ∈ H[X]$, put $$d(x,A) = \inf \{d(x, y) : y ∈ A\}$$ where on the right $d$ ...
3
votes
2answers
301 views

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.
4
votes
1answer
130 views

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 ...
6
votes
1answer
197 views

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 ...
12
votes
3answers
593 views

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 ...
6
votes
1answer
288 views

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 ...
0
votes
1answer
125 views

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 ...
5
votes
2answers
243 views

Polish space in which the interior of each compact set is empty

If anyone could give me an example of Polish space, in which the interior of each compact set is empty? I guess it is trivial, but can't find such an example.
8
votes
2answers
863 views

Compact subset of an open set

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 $B(x,r)\subset A$, ...