3
votes
1answer
18 views

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 ...
4
votes
1answer
98 views

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). ...
3
votes
1answer
84 views

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 ...
0
votes
1answer
26 views

Boolean Closure and Borel sets

Denote the boolean closure of a family of sets $\mathcal S$ by $\mathcal B(\mathcal F)$, then in a metric space it is well known that $\mathcal B(\mathcal F) = \mathcal B(\mathcal G) = \mathcal ...
3
votes
0answers
40 views

On Knaster-Kuratowski fan (aka Cantor's teepee)

As definition of Knaster-Kuratowski fan I take the one on wikipedia. Now, part of the definition of this space is the Cantor set. My question is what particular property of Cantor set we necessary ...
1
vote
1answer
52 views

Is there a name for a topological space $X$ in which Every closed subset $A\subsetneq X$ is contained in a countable union of compact sets

As was recommended for me in here I would like to share the following question with you: Is there a name for a topological space $X$ which satisfies the following condition: Every closed subset ...
2
votes
1answer
48 views

Is there a name for a topological space $X$ in which very closed set is contained in a countable union of compact sets?

Is there a name for a topological space $X$ which satisfies the following condition: Every closed set in $X$ is contained in a countable union of compact sets Does Baire space satisfy this ...
1
vote
1answer
53 views

Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
0
votes
3answers
60 views

How would I show that a set is not dense?

Consider the Cantor set. I want to prove that it is not dense in $\mathbb{R}$. Now it makes sense to me since the Cantor set misses entire open intervals in $\mathbb{R}$. So my idea was to choose some ...
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 ...
1
vote
1answer
72 views

Is any subset of the open ordinal space $[0,\Omega)$ $G_\delta$?

Consider the open ordinal space $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal. Can I say that every subset of $[0,\Omega)$ is $G_\delta$? If yes, does this imply that $[0,\Omega)$ is ...
3
votes
1answer
42 views

are the regular topology on $\mathbb R$ and the product topology on $\omega^\omega$ equivalent?

are the regular topology on $\mathbb R$ and the product topology on $\omega^\omega$ equivalent? By the product topology on $\omega^\omega$ I mean the topology in which an open basis set is a set of ...
1
vote
1answer
69 views

Banach Mazur game - Oxtoby - Thm 6.1

I have asked about this theorem before but found lately that I still don't fully understand its proof. Here are the rules of the game described. A closed interval in $\mathbb{R}$ denoted $I_0$ is ...
1
vote
1answer
50 views

Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that, Thm 6.1: There is a strategy in which is sure to win iff is of first category The game played is this: there is a set ...
2
votes
1answer
41 views

Embedding 2nd countable, zero-dimensional Hausdorff space in the Cantor space

The Cantor Space $2^{\mathbb N}$ is the space of all infinite $0$-$1$-sequences with the metric $d(x,y) = 0$ for $x=y$ or $d(x,y) = 1/k$ where $k$ is the least integer such that $x_k \ne y_k$. Now I ...
0
votes
1answer
60 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
2
votes
1answer
30 views

Oxtoby Thm 5.4 Bernstein sets

I am reading Measure and Category of Oxtoby. I have a question about Theorem 5.4 added below. I think I understand the construction of Bernstein sets, and also the main line of the Proof. My question ...
0
votes
0answers
23 views

When do continuous surjections have Borel sections?

It is known that whenever we have a continuous, surjective map $f\colon X\to Y$ between compact metrisable spaces, there is a Borel (even Baire class $1$) section $g\colon Y\to X$ (so that $f\circ ...
3
votes
2answers
113 views

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 ...
4
votes
1answer
73 views

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.
0
votes
0answers
33 views

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 ...
0
votes
2answers
53 views

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 ...
1
vote
2answers
80 views

Are the perfect-free sets countable?

Let $A$ be subset of $\mathbb{R}$ that contains no nonempty perfect subsets. Is $A$ countable?
2
votes
1answer
60 views

Selection Principles Question /Cantor Set

Given, say a perfect polish space $P$ that contains a Cantor set $C$. Let $\mathcal A$ be $\mathcal P(\mathcal P(C))$. Given a sequence $\langle U_n : n\in\omega\rangle\in{}^\omega\mathcal{A}$, $U_i ...
2
votes
1answer
94 views

Proof that the space of infinite 01-sequences (Cantor-space) is totally disconnected

I want to proof that the space $\{0,1\}^{\mathbb N}$ of infinite binary sequences with the product topology is totally disconnected. I know that this space has a basis consisting of clopen sets and is ...
3
votes
1answer
56 views

An example of a $F_{\sigma\delta}$ subset of $[0,1]$ of measure $1$ which is not $F_\sigma$

I'm trying to understand Borel sets. I am looking for a visual (i.e., constructive) $F_{\sigma\delta}$ subset of $[0,1]$ of measure $1$ which is not $F_\sigma$. Any idea or suggestion would be ...
0
votes
1answer
103 views

The Topologies generated by Borel sets.

If I have a topological space $(X, \tau)$, then we can consider the Borel hierarchy on it. Now the sets from a Borel class itself could be taken as generating a topology. Is something known or what ...
1
vote
2answers
83 views

A subspace of the Cantor space

Is The Cantor space $2^{\omega}$ homeomorphic to the subspace $2^{\omega} - \{(1,1,1,1,1,....)\}$ . If not, what it can be said about this subspace?
4
votes
1answer
128 views

The set of all compact non-empty subsets is perfect

Let $X$ be a perfect Polish space and let $H[X]$ be the set of all non-empty compact subsets of $X$. For $A,B \in H[X]$ define the so called Hausdorff-Distance $$ d_H(A,B) = \max \{ \sup_{x \in X} ...
2
votes
1answer
41 views

Finding a $V$ open s.t. $V^{-1}V \subset U$

I'm trying to understand a sentence in a theorem about Polish groups. I'll write the proof all the way up to the sentence that I'm having trouble with. Theorem: Let $G,H$ be Polish groups and $\phi: ...
1
vote
1answer
100 views

What can we say about closed sets in the Baire space that are neither open nor compact?

I'm trying to figure out what closed subsets in $\omega^{\omega}$ equipped with product topology should look like. It seems to me it's relatively easy to have an idea about compact closed subsets and ...
0
votes
1answer
99 views

Why in a space in which compact sets have empty interior, the closure of open sets is not totally bounded?

I'm trying to understand the proof of the theorem: The Baire space $\mathcal{N}$ is unique up to homeomorphism, non-empty zero-dimentional Polish space for which all compact subsets have empty ...
2
votes
1answer
91 views

Using Minimax Theorem of prove the determinacy of closed games?

In this paper, page 6, Itai Arieli and Yehuda Levy mention briefly using using Minimax Theorem of prove the determinacy of closed game in a more general setting. The minimax theorem they mentioned ...
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 $X$, ...
1
vote
1answer
80 views

The graph of Borel measurable function whose range is a separable metrisable space

If $Y$ is separable and $f : X \to Y$ is Borel measurable, then the graph of $f$ is Borel. On page 14, Lemma 2.3, (iii) of this online note, given, $\{U_n\}_{n \in \Bbb N}$, a basis for the ...
2
votes
1answer
77 views

Question on the proof of a subspace of Polish space is Polish, iff it's a $G_\delta$ set.

Suppose, $X$ is a Polish space, $Y$ is a Polish subspace of $X$. $\{U_n\}_{n \in \Bbb N}$ is a basis of open sets of $X$. Let $A = \{ x \in \overline {Y} : \forall \epsilon \exists {n}(x \in U_n ...
5
votes
2answers
197 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? ...
3
votes
1answer
46 views

Is it possible that the union of a Bernstein set and a singleton isn't a Bernstein set?

Since the construction of a partition of two Bernstein sets is almost identical to that of a partition of three in an uncountable Polish space. It's possible that the union of a Bernstein set and a ...
9
votes
0answers
157 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
2
votes
1answer
159 views

Quotient space and Retractions

I'm trying to learn something about topology and category theory. Let us consider the category of compact Polish spaces. The category contains all quotients of all objects (wikipedia) For an ...
3
votes
1answer
70 views

Why the set of outcomes generated by a fixed strategy of one player in Gale-Stewart game is a perfect set?

In the proof that there is a payoff set $X$ such that the Gale-Stewart game is not determined(see here, Proposition 3.1.). I don't know why $X$, the set of all outcomes generated by a fixed strategy ...
4
votes
2answers
192 views

$\{0,1\}^\mathbb{N}$ is homeomorphic to which subset of $\mathbb{R}$.

In an interview the interviewer asked me the following but I failed to give the answer. $\{0,1\}^\mathbb{N}$ with product topology is homeomorphic to which subset of $\mathbb{R}$? Can anyone give ...
1
vote
1answer
97 views

non-Lebesgue-measurable subsets of Smith-Volterra-Cantor sets

For the Smith-Volterra-Cantor set (or simply SVC) we define an equivalence relation R by making each connected component in SVC an equivalence class. It is easy to see that the collection of all ...
1
vote
0answers
35 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
3
votes
2answers
125 views

Uniform versus product topologies on $[0,1]^\mathbb{N}$, and their Borel $\sigma$-algebras

Let $\tau_U$ and $\tau_P$ be the uniform (i.e. sup-metric) and product topologies on $[0,1]^\mathbb{N}$, respectively. Clearly, these topologies are not the same ($\tau_P$ is separable and $\tau_U$ ...
20
votes
2answers
478 views

What's the difference between rationals and irrationals - topologically?

I know that sets of rational and irrational numbers are quite different. In measure, almost no real number is rational and of course, $\mathrm{card}(\mathbb Q) < \mathrm{card}(\mathbb R \setminus ...
5
votes
0answers
161 views

Baire sets of $X$ possess the required Cartesian product property

Let $X=X_{1}\times X_{2}$ is locally compact space, and define $$E=\{E_{1}\times E_{2}\;|\; E_{i}\; \text{is a Borel set in}\; X_{i}\; ,\; \text{for}\; i=1,2\}$$ Now why the Baire sets of $X$ are in ...
2
votes
2answers
263 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.
18
votes
1answer
545 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
2
votes
1answer
62 views

Why is this set nonempty? (Kechris, If X if comp. metriz., so is K(X))

Source: Kechris, pg 26, Theorem 4.25 If $X$ is completely metrizable, so is $K(X)$. Fix a complete compatible metric $d \leq 1$ on $X$. Let $(K_n)$ be Cauchy in $(K(X),d_H)$, where WLOG we ...