All Questions
Tagged with set-theory gn.general-topology
116 questions with no upvoted or accepted answers
2
votes
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240
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3 questions around the Stone space of the free $\sigma$-algebra on $\omega_1$ free generators
During my studies, I came across several different Stone spaces, e.g.:
(i) The Cantor cube $X=\{0,1\}^{\omega_1}$, which is the Stone space of the free Boolean algebra on $\omega_1$ free generators;
...
- 207
2
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0
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101
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A Baire space with meager projections
Question. Is there a Baire subspace $X$ of a Tychonoff power $M^\kappa$ of some separable metrizable space $M$ such that for any countable subset $A\subset \kappa$ the projection $$X_A=\{x{\...
- 41.9k
2
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371
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Descriptive set theory on $\mathbb{R}^\mathbb{N}$
The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
- 20.5k
1
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70
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A ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz
An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite.
A space $X$ is said to be ...
- 505
1
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0
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84
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Terminology for the property: "Each uncountable disjoint open family is locally countable"
Suppose that a topological space $X$ satisfies the following property
(P): "Each uncountable disjoint open family is locally countable",
where a family $\mathcal U$ of subsets of $X$ is ...
- 505
1
vote
0
answers
88
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Are there results on cardinal function using o-tightness?
Recall that a space $X$ has countable $o$-tightness, if for every family $\mathcal U$ of open sets of $X$
and for each $x \in X$ with $x \in \overline{\bigcup \mathcal U}$, there exists a countable ...
- 621
1
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0
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155
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$f:Y\to X$ continuous with $f^{-1}(x)$ compact for $x\in X$, does there exist a Borel measurable map $g:X\to Y$?
Let $X,Y$ be Polish, metric spaces. $f:Y\to X$ is a continuous, surjective map and for any $x\in X$, $f^{-1}(x)\subset Y$ is compact. Is it true that there is a injective, Borel measurable map $g:X \...
1
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0
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289
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About Whitehead's problem
Hi I am new to proofs of consistency and independence with ZFC of some claims. I have read "The uses of set theory" by Judith Roitman, in that article it is mentioned that the Whitehead ...
- 561
1
vote
0
answers
102
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Functions preserving almost disjoint of partitions
A collection $\mathcal{A}\subseteq \omega^\omega$
is almost disjoint iff
$\bigcap_{X\in \mathcal{A}}X^{-1}(j)$ is finite for all $j\in\omega$.
A function $\Gamma:2^\omega\rightarrow 2^\omega$ is
...
- 909
1
vote
0
answers
280
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Comparing two $\sigma$-algebras
Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$.
Q. For which ...
- 4,058
1
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0
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126
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Partition refinement of a clopen covering in $\Box (\omega+1)^\omega$
Consider $\omega+1$ with the interval topology, that is $U\subseteq (\omega+1)$ is open if and only if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite.
We write $(\omega+1)^\omega$ for the ...
- 48.9k
1
vote
0
answers
321
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Type I subspaces of the Stone Cech compactification of $\omega$
EDIT: I found a construction, see below. I decided not to delete the question in case someone is interested.
A space $X$ is of Type I if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ ...
- 1,855
1
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0
answers
150
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Follow up question on the measure of the difference between a partial selector and a selector...
This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...
In Kharazishvili's "Nonmeasurable Sets and ...
- 463
0
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0
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94
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Is the space of affine continuous functions a Baire space
Let $\Omega$ be a compact convex set in q linear normed space. Let $A(\Omega)$ be the space of affine continuous real-valued functions. My question is whether the space $A(\Omega)$ is a Baire space? ...
- 675
0
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162
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A ``1-soft'' improvement of the Parovichenko theorem
This is a ``1-soft'' modification of this problem. We start with the necessary definitions.
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called 1-soft if for any ...
- 41.9k
0
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0
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643
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A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074