2
votes
1answer
55 views

Is it true that $A$ is scattered?

Let $X$ be a (Hausdorff) topological space and for each ordinal $\alpha$ denote by $X^{(\alpha)}$ the $\alpha$th derivative of $X$ by the Cantor-Bendixson derivation (i.e., define transfinitely: ...
2
votes
1answer
33 views

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets? (or equivalently, that $\mathbb R$ contains no more then $\mathfrak c$ $G_\sigma$ sets? The more general ...
2
votes
0answers
32 views

The principle $S_1(\mathcal O,\mathcal O)$ versus the game $G_1(\mathcal O,\mathcal O)$

Given a topological space $X$, Let $\mathcal O$, denote the set of all open covers of $X$. We say that a space $X$ satisfies $S_1(\mathcal O,\mathcal O)$, if for every sequence of open covers $\{ ...
2
votes
1answer
58 views

Suppose $X$ is a Hausdorff Lindelöf scattered space. Is $\xi(X)$ a successor ordinal?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
0
votes
0answers
22 views

reverse-reverse of Michael selection theorem

Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) ...
2
votes
0answers
20 views

Strong Choquet preimage implies strong Choquet?

Recall that a strong Choquet space is one where player II has a winning strategy in the game where two players take turns: player I chooses an open set and a point inside, then player II chooses a ...
0
votes
1answer
39 views

The Cantor set and it's connection to t(C_p(X))

I am reading the following proof (from page 124 here) which is a part of a larger proof for the statement: Exercise 151: Show that there exist spaces $X$ and $Y$ such that, $t(C_p(X)) = \omega$ and ...
1
vote
2answers
58 views

Homeomorphism between $\prod X_i$ and the Cantor set

I've been trying to find a homeomorphism $$\Phi:\prod_{i=1}^\infty X_i \to \{ 0, 1\}^\mathbb{N},$$ where each $X_i$ is a finite set with at least two elements, but have been unable to. Things I've ...
3
votes
0answers
24 views

How to show: a non-scattered Čech-complete space, contains a compact subspace which can be continuously mapped onto the Cantor-set

It is written in this article that "a non-scattered Čech-complete space, contains a compact subspace which can be continuously mapped onto the Cantor-set". Can anyone explain this? I mean, if, $X$ is ...
3
votes
1answer
66 views

Banach-Mazur Game: Proof about winning strategies

I have to hold a presentation about the Banach-Mazur-Game to undergraduates this week. It should all stay very simple, so I will mainly only talk about the "original" Banach-Mazur Game on ...
0
votes
1answer
54 views

Infinite product space (TOPOLOGY)

I am asking about the infinite product of $\{0,1\}$. That is, $\{0,1\}^\Bbb N$ is the space of all infinite sequence of $0_s$ and $1_s$, how collection of all finite sequences of $0_s$ and $1_s$ ...
1
vote
0answers
22 views

The game $G(K,X)$

In Telgarsky - Topological games, in page 246, the following game $G(K,X)$ is described: There are given a space $X$ and a class $K$ of spaces such that $Y \in K \Rightarrow \mathcal F(Y) \subset K$. ...
17
votes
1answer
449 views

Why do we study Polish spaces?

In descriptive set theory, a lot of space is devoted to properties of Polish spaces. (A Polish space is a topological space, which is separable and completely metrizable.) I would like to know why ...
1
vote
2answers
65 views

Can we construct from $[0,\omega_1)$ a space which is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

I have asked in here a question which tured out to make no sense. I think I have found the confusion and would like to try and rephrase my question: Let $E$ be a topological space, $q \in E$. ...
1
vote
0answers
27 views

Can we say that $[0,\omega_1]$ is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

Let $E$ be a topological space, $q \in E$. The neighbourhood point game $G_{np}(q,E)$, is defined as follows. It is played by two players, ONE and TWO.In the n's step $n \in \omega$, ONE chooses ...
3
votes
1answer
43 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
173 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
2answers
118 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
34 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
59 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
56 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
54 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
67 views

Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
0
votes
3answers
77 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
92 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
43 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
90 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 ...
2
votes
1answer
60 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
57 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
70 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 ...
3
votes
1answer
35 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 ...
1
vote
0answers
25 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
127 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
78 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
38 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
55 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
87 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
64 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
110 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
59 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
111 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
131 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
42 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
109 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
103 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
93 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
85 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 ...