All Questions
Tagged with set-theory gn.general-topology
433 questions
14
votes
3
answers
1k
views
Order homomorphism functions on $\omega_1$
I posted the following question more than two years ago on MO (and then reposted on MSE), but the answer remains incomplete, so I thought I would rephrase it a bit (to make the statement clearer) and ...
- 1,375
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
14
votes
0
answers
543
views
Small cardinals related to topological convergence
Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-...
- 4,413
13
votes
3
answers
820
views
Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
This question is related to another one that I asked two days ago.
Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
the following two properties?
The ...
- 1,396
13
votes
1
answer
639
views
$T_2$-spaces where all non-empty open sets are homeomorphic
We say that a $T_2$-space $(X,\tau)$ has homeomorphic open sets if every non-empty open set $U\subseteq X$ endowed with the subspace topology is homeomorphic to $(X,\tau)$.
The rationals with the ...
- 48.9k
13
votes
1
answer
558
views
Idempotent ultrafilters and the Rudin-Keisler ordering
Short version: what can we say about the place of idempotent ultrafilters in the Rudin-Keisler ordering?
Longer version:
If $U$, $V$ are (nonprincipal) ultrafilters on $\omega$, then we write $U\ge_{...
- 20.5k
13
votes
2
answers
690
views
How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?
I can see that results in both ...
- 233
13
votes
1
answer
674
views
Avoiding countable subgroups of a group homeomorphic to the Cantor space
Update: Further work with Adam (who answers below) and Piotr led to a rather satisfactory result about the problem that motivated the problem below, see our recent paper
The Haar Measure Problem. In ...
- 3,104
13
votes
1
answer
1k
views
Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhaps some other set?
Is every sigma-algebra the Borel algebra of a topology?
inspires the present question which asks for less.
Question: Given a $\sigma$-algebra $\mathcal A$ on a set $X$, does there exist a topology $\...
- 17.6k
13
votes
0
answers
421
views
A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?
Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
- 41.9k
12
votes
1
answer
516
views
Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?
Is $[0,1]$ a disjoint union of $\aleph_1$ compact subsets with empty interior?
The answer is obviously yes assuming the continuum hypothesis. Also, by Baire's lemma, the answer is negative if one ...
- 3,146
12
votes
1
answer
635
views
Ultrafilter subtraction and "zero"
This is related to a couple recent MO/MSE questions of mine, namely 1,2. Belatedly, I've tweaked this post to remove an overly-ambitious secondary question; see the edit history if interested.
Let $\...
- 20.5k
12
votes
3
answers
1k
views
If Q is a subset of the plane of size less than continuum, then does every closed F in Q extend to a closed connected G in the plane with the same trace on Q? (Or is this independent of ZFC?)
This question arises in connection with this MO
question
and especially with Sergei Ivanov's wonderful
answer,
which showed that for any countable set
$Q\subset\mathbb{R}^2$ and every closed set $F\...
- 236k
12
votes
1
answer
447
views
Discrete subsets in the topology of pointwise convergence vs. metrisability
While reading Arkhangel'skii's Topological function spaces, I encountered an unexpected application of Martin's Axiom. This is Theorem II.5.20:
Assume $\mathsf{MA}+\neg \mathsf{CH}$. Let $X$ be a ...
- 11.3k
12
votes
1
answer
316
views
A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube
I am looking for a reference of the following (true) fact:
Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
- 41.9k
12
votes
1
answer
582
views
Is a locally finite union of $G_\delta$-sets a $G_\delta$-set?
Problem. Let $\mathcal F$ be a locally finite (or even discrete) family of (closed) $G_\delta$-sets in a topological space $X$. Is the union $\cup\mathcal F$ a $G_\delta$-set in $X$?
Remark. The ...
- 41.9k
12
votes
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
12
votes
0
answers
386
views
L-spaces without convergent sequences
An L-space is a regular hereditarily Lindelof space which is not hereditarily separable. Consistent examples of L-spaces are relatively easy to come by (for example, Suslin Lines), but the first ...
- 4,413
12
votes
0
answers
219
views
Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?
Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.
Let
$$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$
for all $n\in\mathbb{N}$.
Then since $C_{n}$ is a ...
11
votes
5
answers
1k
views
Confusion over a point in basic category theory
"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
- 1,207
11
votes
3
answers
890
views
Structure theorems for compact sets of rationals
Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in ...
- 1,882
11
votes
2
answers
605
views
Example of an uncountable scattered space with some properties
This might be an easy question, maybe the example I'm looking for is common knowledge. As always, recall that a topological space $X$ is scattered if and only if every non-empty subset $Y$ of $X$ ...
- 674
11
votes
1
answer
2k
views
Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra?
Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$?
It is not true for arbitrary Boolean algebras, see the comments, or see my MathSE ...
- 263
11
votes
2
answers
725
views
Is a Borel image of a Polish space analytic?
A topological space $X$ is called analytic if it is a continuous image of a Polish space, i.e., the image of a Polish space $P$ under a continuous surjective map $f:P\to X$.
We say that a topological ...
- 41.9k
11
votes
1
answer
769
views
Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?
Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition:
The meager sets are sets which are ...
- 1,416
11
votes
1
answer
548
views
Non meager rectangle
Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?
- 9,641
11
votes
1
answer
408
views
The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
- 41.9k
11
votes
2
answers
721
views
Inconsistency and workaday independence.
Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
- 17.6k
11
votes
1
answer
704
views
Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$
We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions.
One can generalize the definition above by taking pointwise limit of ...
- 623
11
votes
1
answer
799
views
Restrictions of null/meager ideal
Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
- 9,641
11
votes
0
answers
172
views
Can the nowhere dense sets be more complicated than the meager sets?
Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
- 18.6k
11
votes
0
answers
322
views
Does any real function have a Lipschitzian restriction on $D$?
Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
- 4,074
11
votes
0
answers
273
views
A ZFC-example of a countably compact paratopological group which is not a topological group
Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group?
(The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...
- 4,535
10
votes
2
answers
315
views
Limits of rearranged sequences along ultrafilters
Suppose that a bounded sequence of real numbers $s_i$ ($i\in\omega$) has a limit $\alpha$ along some ultrafilter $\mu_1\in \beta{\Bbb N}\setminus{\Bbb N}$. Then given another ultrafilter $\mu_2\in \...
- 17.6k
10
votes
2
answers
244
views
Minimal refinements of open covers of $T_2$-spaces
Let $(X,\tau)$ be a topological space. We say ${\cal U}\subseteq \tau$ is an open cover if
$\bigcup {\cal U} = X$, and
$X\notin {\cal U}$.
${\cal U}$ is minimal if for all $U_0\in {\cal U}$ we have ...
- 48.9k
10
votes
1
answer
326
views
What is known about topological groups of countable spread in ZFC?
A topological space has countable spread if every discrete subspace is at most countable.
By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X$...
- 41.9k
10
votes
2
answers
750
views
Is there a compact space with no countably generated dense subspace?
This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give ...
- 11.5k
10
votes
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
10
votes
2
answers
426
views
Two questions about the "grasp" cardinal function
For a topology $\mathcal{T}$ on a set $S$, where $\mathcal{T}$ does not have a finite base, I define the grasp $g(\mathcal{T})$ to be the least infinite cardinal $\kappa$ such that $\mathcal{T}$ has a ...
- 337
10
votes
1
answer
460
views
An incomplete characterisation of the Euclidean line?
We say that a metric space $(X, d)$ is a Banakh space if for every $\rho \in \mathbb{R}_{> 0}$ and every $x \in X$, there are
$a,b \in X$ such that $\{y \in X \, \vert \, d(x, y) = \rho\} = \{a, b\}...
- 7,893
10
votes
1
answer
354
views
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
10
votes
1
answer
417
views
A variant of the Moore-Mrowka problem
A space $X$ is said to be sequential if whenever $A \subset X$ is not closed then $A$ contains a sequence converging to a point outside of $A$.
A space $X$ is said to have countable tightness if for ...
- 4,413
10
votes
1
answer
514
views
Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$
Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.
...
- 41.9k
10
votes
2
answers
363
views
Source on smooth equivalence relations under continuous reducibility?
This question was asked and bountied at MSE, but received no answer.
In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
- 20.5k
10
votes
1
answer
350
views
What is the smallest density of a metrizable space without countable separation?
A Tychonoff space $X$ is defined to have countable separation if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\...
- 41.9k
10
votes
0
answers
242
views
Arhangel'skii's problem revisited
One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
- 4,413
10
votes
0
answers
323
views
Determinacy coincidence at $\omega_1$: is CH needed?
This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
- 20.5k
10
votes
0
answers
370
views
+400
Extending models of topological set theory
$\mathsf{GPK_\infty^+}$ is an alternative set theory in which we have comprehension for formulas which are positive in a certain sense; see the SEP article for more detail (or this MO post, which ...
- 20.5k
10
votes
0
answers
293
views
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; ...
- 20.5k
10
votes
0
answers
498
views
Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?
Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals:
$\mathfrak p$ is the ...
- 1,101