# Tagged Questions

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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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 $$d_H(A,B) = \max \{ \sup_{x \in X} ... 1answer 42 views ### Finding a V open s.t. V^{-1}V \subset U I'm trying to understand a sentence in a theorem about Polish groups. I'll write the proof all the way up to the sentence that I'm having trouble with. Theorem: Let G,H be Polish groups and \phi: ... 1answer 108 views ### What can we say about closed sets in the Baire space that are neither open nor compact? I'm trying to figure out what closed subsets in \omega^{\omega} equipped with product topology should look like. It seems to me it's relatively easy to have an idea about compact closed subsets and ... 1answer 103 views ### Why in a space in which compact sets have empty interior, the closure of open sets is not totally bounded? I'm trying to understand the proof of the theorem: The Baire space \mathcal{N} is unique up to homeomorphism, non-empty zero-dimentional Polish space for which all compact subsets have empty ... 1answer 93 views ### Using Minimax Theorem of prove the determinacy of closed games? In this paper, page 6, Itai Arieli and Yehuda Levy mention briefly using using Minimax Theorem of prove the determinacy of closed game in a more general setting. The minimax theorem they mentioned ... 3answers 76 views ### Open neighborhoods of a G_\delta set This may have a simple answer, but I couldn't find it so far either in textbooks or in math.stackexchange. Let X be a metric space, and$$A=\bigcap^\infty_{n=1}A_n a $G_\delta$ subset of $X$, ...
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 ...