All Questions
Tagged with set-theory gn.general-topology
433 questions
1
vote
2
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273
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A Borel perfectly everywhere surjective function on the Cantor set
Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set ...
- 3,115
2
votes
0
answers
162
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Banach–Mazur game and mappings
The Banach-Mazur game on a nonempty space $X$ is defined as follows: two players, $I$ and $II$, alternately choose nonempty open sets
\begin{matrix}
I & U_0 && U_1 && \cdots ...
- 115
5
votes
2
answers
314
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Self-homeomorphism of Stone-Čech boundary with an isolated fixed point
$\DeclareMathOperator\bso{\beta^*\!\omega}\DeclareMathOperator\Homeo{Homeo}$Let $\bso$ be the complement of the countable discrete space $\omega$ in its Stone-Čech compactification $\beta\omega$ (some ...
- 63.9k
1
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0
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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
3
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1
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510
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Can Tychonoff's theorem be applied to topological spaces generated by program output in ZFC?
I am confused about an issue in set theory.
Tychonoff's theorem says that "an arbitrary product of compact topological spaces is compact". We often talk of an index set $I$ and then for each ...
- 41.4k
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
7
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1
answer
397
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A set theoretic question arising from trying to understand a sheaf cohomology question
I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
- 38.6k
15
votes
2
answers
341
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Do we need full choice to "efficiently" use (sub)bases?
This question was previously asked and bountied at MSE without success.
Suppose $(X,\tau)$ is a topological space, $B$ is a base for $\tau$, and $U\in \tau$ is an open set. Consider the following two ...
- 21.2k
2
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1
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188
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The Tychonov cube $X^X$ of a compact space $X$ is a compact semigroup with the composition operation
Reading a book about Ramsey theory this is the first example of a compact (semitopological) semigroup, which is a nonempty semigroup S with compact Hausdorff topology for which $x \mapsto x*s$ is a
...
- 59
1
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1
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143
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Is the Rudin-Keisler ordering a continuous relation?
If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is continuous (from $X$ to $Y$) if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\...
- 48.9k
0
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1
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103
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(Seeking Definition) What Does it Mean for a Space to have Rim-Type $\alpha$? Or the 'derivative' of a Countable Set?
I've encountered a definition in several papers, but literally none of them define the term. They all instead reference a book by Menger that has never been printed in English. The term is "rim-...
- 439
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0
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190
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What is the smallest number of nowhere dense affine subsets covering a topological group?
$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$.
Given a non-discrete topological ...
- 41.9k
5
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143
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Two cardinal characteristics of the continuum, related to the Bohr topology on integers
For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in ...
- 41.9k
5
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0
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170
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Who should be attributed for the definition of almost disjoint families of true cardinality $\mathfrak{c}$?
A family $\mathcal{A}$ of infinite subsets of $\omega$ is called
almost disjoint if for any two distinct sets $a, b \in \mathcal{A}$, the intersection $a\cap b$ is finite.
An almost disjoint family $\...
- 105
4
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1
answer
718
views
Is every element of $\omega_1$ the rank of some Borel set?
It is well known that we can obtain the $\sigma$-algebra of Borel subsets of $2^{\omega}$ in the following way: Let $B_0$ be the collection of all open subsets of $2^{\omega}$. For $\alpha=\beta+1$, ...
- 1,799
8
votes
1
answer
183
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Are all monotonically normal manifolds of dimension at least two metrizable?
Alan Dow and Frank Tall recently proved the consistency of the statement Every hereditarily normal manifold of dimension at least two is metrizable.
See: Dow, Alan; Tall, Franklin D., Hereditarily ...
- 4,413
1
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1
answer
108
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The cardinal of the closure of a set in a topological space
Suppose $E$ is a Hausdorff topological space, $A$ is a subset of $E$. Card(A) is the cardinal of $A$. $\bar{A}$ is the closure of $A$. Do we have
$Card(\bar{A})\le 2^{2^{Card(A)}}$.
- 11
6
votes
1
answer
205
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Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$
It is a well known fact that if $\mathcal{F}$ is a non-principal ultrafilter on $\omega$, then the set $\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$ (conflating binary strings with subsets of $\...
- 12.5k
8
votes
3
answers
937
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BCT equivalent to DC
Do you know where I can find proof of equivalence Baire Category Theorem and DC (Axiom of Dependent Choice)? It is well known fact but I can't find appropriate literature with the proof.
- 143
17
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1
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429
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Axiom of Countable Choice and meager sets
Let us recall that the Axiom of Countable Choice (denoted by ACC) says that the countable product $\prod_{n\in\omega}X_n$ of nonempty sets $X_n$ is nonempty.
It is easy to see that ACC implies that ...
- 41.9k
0
votes
1
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170
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P-filter property?
Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup_i ...
- 1,101
7
votes
0
answers
221
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adding one point from the Stone-Cech compactification
Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
- 71
6
votes
2
answers
1k
views
Foundational results dependent on/equivalent to the continuum hypothesis or its negation?
I remember at a certain point early in my mathematical studies learning that the Axiom of Choice is equivalent to the following statement on Cartesian products:
If $\{ X_i \}_{i \in I}$ is any ...
- 263
10
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1
answer
333
views
Possible cardinalities of the remainders of compactifications of $\Bbb R$
With the usual topology on $\Bbb R$, a compactification $\mathrm{id}_{\Bbb R}:\Bbb R\to v\Bbb R$ can have a remainder $v\Bbb R \setminus \Bbb R$ of cardinality $1,2, 2^{\aleph_0}=\mathfrak c,$ or $2^{\...
- 337
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281
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Is there a condensation from $\aleph_1^{\aleph_0}$ onto a metrizable compact space?
Is there a condensation (continuous bijective mapping) from $D^{\aleph_0}$ onto a metrizable compact space ?
$D$ - discrete space of cardinality $\aleph_1$.
CH implies it is a positive answer. In ...
- 1,101
5
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0
answers
265
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Uncountable subspaces of the real line
Definition 1. A topological space $\langle X, \tau \rangle$ is meager if $X = \bigcup_{n \in \omega}A_n$, where each $A_n$ is nowhere dense in $\langle X, \tau \rangle$.
Definition 2. A topological ...
- 115
1
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0
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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
9
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1
answer
831
views
Baire category theorem for uncountable unions
Any compact Hausdorff space $X$ is a Baire space:
if the set $X$ is a meager set (meaning a countable union of nowhere dense subsets,
also known as a set of first category),
then $X$ is empty.
I am ...
- 37.8k
18
votes
0
answers
370
views
Čech functions and the axiom of choice
A Čech closure function on $\omega$ is a function $\varphi:\mathcal P(\omega)\to\mathcal P(\omega)$ such that (i) $X\subseteq\varphi(X)$ for all $X\subseteq\omega$, (ii) $\varphi(\emptyset)=\emptyset$,...
- 13.4k
4
votes
0
answers
156
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Does $\mathbb{R}$ have a partite subbase?
If $X\neq \varnothing$ is a set we say that ${\frak P} \subseteq {\cal P}(X)$ is a partition of $X$ if
$\bigcup{\frak P} = X$, and
$P\neq Q \in {\frak P} \implies P\cap Q = \varnothing$.
Let $H = (V,...
- 48.9k
1
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2
answers
176
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Connected Hausdorff spaces with large collection of disjoint open sets
Is there for every infinite cardinal $\kappa$ a connected Hausdorff space $(X,\tau)$ with $|X| = \kappa$ and a collection ${\cal D}$ of mutually disjoint open sets with $|{\cal D}| = \kappa$?
- 48.9k
1
vote
2
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223
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Covering dimension of uncountable union of compact spaces
It is well-known that the (covering) dimension of countable union of compact spaces is the superium dimension of these spaces. I would like to understand certain uncountable union of compact spaces as ...
- 675
1
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1
answer
278
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Borel hierarchy and tail sets
Let $A$ be a finite set, and let $A^\infty$ be the set of all sequences $(a_n)_{n=1}^\infty$ of elements of $A$.
A set $B \subseteq A^\infty$ is a tail set if for every two sequences $\vec a, \vec b \...
- 745
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0
answers
241
views
Is there a characterization of the class of first-order formulas that are closed in every compact Hausdorff structure?
Fix a relational language $\mathcal{L}$. (I don't think relational really matters that much but I don't want to worry about it.) A topological $\mathcal{L}$-structure is an $\mathcal{L}$-structure $M$ ...
- 12.5k
1
vote
1
answer
136
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A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$
Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech ...
- 41.9k
5
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1
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524
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A topologically transitive dynamical system without dense orbits
By a dynamical system I understand a pair $(K,G)$ consisting a compact Hausdorff space and a subgroup $G$ of the homeomorphism group of $K$.
We say that a dynamical system $(K,G)$
$\bullet$ is ...
- 41.9k
6
votes
3
answers
472
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Spaces with unique endomorphism monoids
If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$.
We ...
- 48.9k
5
votes
4
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753
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Questions about existence of injections between infinite sets and the sets of all infinite topologies on them
1) If $X$ is an infinite set and $T_X$ the set of all infinite topologies on $X$ is it in general true that there is no injection $f_T:T_X \to X$?
2) What conditions on $X$ assure an injection (if ...
user153451
2
votes
1
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348
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On the hereditary Lindelof topological spaces
I received the following interesting point in (1). I could not find any reference or clear proof. Any suggestion?
Theorem. A topological space $X$ is hereditary Lindelof if and only if for any ...
- 4,058
10
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1
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354
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Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?
Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures.
Consider the endomorphism $\hat{\Phi}$ ...
- 63.9k
14
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2
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502
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Near permutation $n\mapsto n+1$ not conjugate to its inverse on the Stone-Čech remainder?
Let $\beta\omega$ be the Stone-Čech compactification of the discrete infinite countable space $\omega$, and $\beta^*\omega=\beta\omega\smallsetminus \omega$ is the Stone-Čech remainder.
The map $j:n\...
- 63.9k
14
votes
0
answers
851
views
Cardinality of the set of continuous functions
Suppose $(X,\tau)$ and $(Y,\sigma)$ are topological spaces. Let $F(X,Y)$ be the set of continuous functions $X\rightarrow Y$. I want to compute the cardinality of $F(X,Y)$. It depends not only on ...
- 12.3k
10
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0
answers
293
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Undetermined Banach-Mazur games: beyond DC
This question is a follow-up to this one; see that question for the definition of Banach-Mazur games. There James Hanson showed that ZF+DC proves that there is an undetermined Banach-Mazur game; ...
- 21.2k
22
votes
1
answer
754
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Undetermined Banach-Mazur games in ZF?
This question was previously asked and bountied on MSE, with no response. This MO question is related, but is also unanswered and the comments do not appear to address this question.
Given a ...
- 21.2k
2
votes
1
answer
135
views
Compactifications with remainder $[0,\omega_1]$ and convergent sequences
Is the following statement consistent?
$(\star)$ Let $K$ be a separable compact Hausdorff space containing the space $[0,\omega_1]$ so that the complement $K\setminus[0,\omega_1]$ is discrete. Then ...
- 41.9k
0
votes
0
answers
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
7
votes
1
answer
366
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The diamond principle for functors
Let $F:\mathbf{Comp}\to\mathbf{Set}$ be a continuous functor from the category of compact Hausdorff spaces to the category of sets such that $|Fn|\le\mathfrak c$ for any finite ordinal $n$. The ...
- 41.9k
9
votes
0
answers
367
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A $\mathsf{ZF}$ example of two Baire spaces whose product is not Baire?
Motivated by this question I'm looking for a pair of Baire topological spaces whose product is not Baire and whose construction does not need the axiom of choice.
The example of such spaces I'm ...
- 3,011
5
votes
0
answers
132
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Infinite connected $T_2$-space with fixed-cardinality fibers
What is an example of a connected $T_2$-space $(X,\tau)$ with $X$ infinite and the following property?
If $\alpha \leq |X|$ is a non-empty cardinal, then there is a continuous map $f:X\to X$ such ...
- 48.9k