All Questions
Tagged with set-theory gn.general-topology
433 questions
155
votes
4
answers
18k
views
Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
- 39k
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
15
votes
2
answers
965
views
$\mathfrak{ufo}$: An unidentified combinatorial cardinal characteristic of the continuum?
An ultrafilter ornament is a chain of free filters on $\mathbb{N}$ that are not ultrafilters, whose union is an ultrafilter.
Let $\mathfrak{ufo}$ be the minimal cardinality of
an ultrafilter ...
- 3,104
7
votes
1
answer
389
views
References for higher descriptive set theory surveys
A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
- 3,104
6
votes
3
answers
655
views
When does the generalized Cantor space embed in a $\kappa$-compact space
The generalized Cantor space is the space $2^\kappa$, with basic open sets
$$
[\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\},
$$
for $\sigma\in 2^{<\kappa}$.
A space is $\kappa$-compact if ...
- 3,104
2
votes
1
answer
197
views
If $S$ is non-stationary in $[k]^{\omega}$ is there a choice-function on $S$ with bounded fibers?
Fodor's Lemma : When $k$ is a regular uncountable cardinal, and $T$ is a stationary subset of $k$, any regressive $f:T\to k$ has a fiber which is stationary in $k$. Corollary: $T$ is stationary in $...
- 337
5
votes
1
answer
600
views
When is the generalized Cantor space $\kappa$-compact?
My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references.
The generalized Cantor space is the space $2^\kappa$, with basic open ...
- 3,104
15
votes
1
answer
521
views
Is there a locally compact, $\omega_1$-compact, not $\sigma$-countably compact space of size $\aleph_1$?
There are old ZFC examples due to Eric van Douwen that satisfy all the properties in the title, except that they are of cardinality $2^{\aleph_0}$, so the answer to the title question is YES if the ...
- 291
3
votes
1
answer
309
views
Invariants of category in Polish spaces
Consider the invariants appearing in the Cichoń's diagram: $add(\mathcal I)$, $cov(\mathcal I)$, $non(\mathcal I)$, $cof(\mathcal I)$, where $\mathcal I$ is either the ideal of null sets for the ...
- 1,688
7
votes
0
answers
171
views
Are there always large discrete families of normal measures?
Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The ...
- 2,389
5
votes
1
answer
363
views
all subsets borel
Assume Martin's axiom plus $\neg CH$. It is well known, via almost disjoint forcing, that every set of reals of size less than continuum is an example of a metric space whose subsets are all ...
- 51
6
votes
1
answer
223
views
Minimal Hausdorff topologies compatible with a bunch of functions
Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
- 48.9k
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 ...
6
votes
2
answers
309
views
The role of the index set in the product of uncountably many topological spaces
Let $\langle X_i,\mathcal{T}_i \rangle_{i\in I}$ be a family of topological spaces. Consider $X=\prod_{i\in I} X_i$ with product topology.
Question. Is there a topological property that holds in $...
- 1,872
19
votes
0
answers
703
views
The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?
For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
- 3,104
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
21
votes
3
answers
610
views
Which partitions of $[0,1]$ are collection of level sets of a real continuous function?
Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
- 1,014
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
7
votes
0
answers
266
views
Remote points in $\beta X$
It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
- 79
15
votes
1
answer
525
views
A problem of Keisler and Tarski
The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
- 9,631
15
votes
3
answers
717
views
Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?
This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...
- 236k
3
votes
1
answer
294
views
Implications between different covering properties of spaces
Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set,...
- 48.9k
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
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 $A \times B \subseteq G$?
- 9,631
5
votes
3
answers
584
views
The size of Lindelof space
Question. Suppose that $X$ is a Lindelof space such
that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?
user74542
32
votes
1
answer
2k
views
Bidi: A new cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...
- 3,104
16
votes
1
answer
607
views
The dominating number $\mathfrak{d}$ and convergent sequences
All spaces considered below are compact Hausdorff.
If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...
- 1,151
1
vote
0
answers
126
views
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
3
votes
2
answers
294
views
Extending hyperconnected spaces
A hyperconnected space is a topological space such that every two non-empty open sets have non-empty intersection. Let's call a space $(X,\cal{T})$ maximally hyperconnected if it is hyperconnected and ...
user70876
2
votes
1
answer
208
views
Does the countable $\sigma$-product of a separable Hilbert space have a first countable topology?
Let $\mathcal{l}^2$ be "the" separable real infinite dimensional hilbert space, e.g. the space of square-summable sequences of real numbers.
Let $\Box^{\mathbb{N}}\mathcal{l}^2$ be the countable ...
- 21
7
votes
3
answers
528
views
Images of $\{0,1\}^\kappa$
Is there a compact topological space $(X,\tau)$ such that for no cardinal $\kappa$ there is a surjective continuous map $e:\{0,1\}^\kappa \to X$?
(We assume that $\{0,1\}$ is endowed with the ...
- 48.9k
6
votes
3
answers
1k
views
Borel cross section
It is known from metric space topology that a closed equivalence relation on a Polish space has either countably many or $\mathfrak{c}$ many equivalence classes.
A short elementary proof is given in ...
- 1,704
8
votes
1
answer
288
views
Translates of meager sets
Does there exist a meager set of reals M such that every meager set can be covered by countably many translates of M? This is the category analogue of the following.
- 9,631
3
votes
1
answer
338
views
$\text{Cont}(X,X)$ and $\neg\mathsf{GCH}$
For a topological space $(X,\tau)$ let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$. Is it consistent that there a space $(X,\tau)$ such that $$|X| < |\text{Cont}(X,X)| < ...
- 48.9k
2
votes
4
answers
535
views
Topological spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$
Let $(X,\tau)$ be a topological space. Let $\text{Cont}(X,X)$ denote the set of continuous functions $f:X\to X$.
What can be said about spaces $(X,\tau)$ where $|\text{Cont}(X,X)| = |X|$? For ...
- 48.9k
2
votes
1
answer
171
views
Closed sets in ordinal spaces
I'm studying the ordinal space $[0,\kappa[$ where $\kappa\neq \omega$ is a cardinal of countable cofinality and I want to know why there are in $[0,\kappa[$ two disjoint closed sets of cardinality $\...
- 225
4
votes
5
answers
1k
views
A generalized diagonal?
A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It ...
4
votes
1
answer
281
views
CCC Forcing and $\omega_1$ conditions
I have a question about the proof of the Lemma 7.2 in the paper
I. Juhász, P. Koszmider and L. Soukup,
A first countable, initially $\omega_{1}$-compact but non-compact space,
Topology and its ...
- 627
9
votes
0
answers
624
views
Two questions about universally measurable sets
I have two questions about universally measurable sets:
(1) Is there a universally measurable set of reals which does not have the Baire property?
(2) Is there a universally measurable set of reals ...
- 9,631
2
votes
1
answer
334
views
How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?
To make the question more precise:
We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$.
Let $\mathcal{C}$ be ...
- 48.9k
6
votes
2
answers
510
views
Gorelic's Forcing for large Lindelöf spaces with points $G_\delta$
I am trying to understand a step for proving that there exists large Hausdorff Lindelöf Spaces with points $G_\delta$ using forcing. I am following Isaac Gorelic's "The Baire Category And Forcing ...
user46747
6
votes
0
answers
105
views
Large discrete subspaces in spaces of separately continuous functions
For topological spaces $X,Y,Z$ let $SC_p(X\times Y,Z)$ be the space of separately continuous functions $f:X\times Y\to Z$ endowed with the topology of pointwise convergence.
It is easy to see that ...
- 41.9k
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
6
votes
2
answers
582
views
If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
The answer is negative, and in the interests of self-contained ...
- 2,111
8
votes
2
answers
577
views
Ultracoproducts and Cartesian products
Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...
- 1,786
5
votes
2
answers
444
views
non-Borel set which intersects every compact in a Borel set
I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...
- 1,704
5
votes
1
answer
223
views
A realcompact analogue of the Baire category theorem
Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...
- 3,041
8
votes
1
answer
571
views
Is $\ell^\infty$ Polishable?
Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
- 3,115
3
votes
1
answer
202
views
Ultrafilters of weight $\aleph_2$ in Sacks model
It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model P-...
9
votes
1
answer
531
views
Existence of infinite groups that are too reluctant to be topological
With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
- 1,145