All Questions
Tagged with gn.general-topology set-theory
433 questions
3
votes
0
answers
209
views
Compactification of Tychonoff spaces without full axiom of choice
If $X$ is a Tychonoff space, then using the Tychonoff theorem and thus the full axiom of choice, it follows that $X$ admits a Hausdorff compactification.
My question is : what remains true if we do ...
- 506
5
votes
1
answer
528
views
Base zero-dimensional spaces
Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
- 41.8k
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 $...
- 41.8k
33
votes
1
answer
2k
views
Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
- 4,597
6
votes
1
answer
772
views
A ridiculous combinatorial cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the ...
- 3,104
-1
votes
1
answer
122
views
Injective choice function for non-separable $T_2$-spaces
For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ be the collection of all subsets of $X$ of cardinality $\kappa$.
I was looking for $T_2$-spaces $(X,\tau)$ with the property that
$(P)$ ...
- 48.9k
4
votes
0
answers
122
views
Completely I-non-measurable unions in Polish spaces
Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
- 4,535
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
7
votes
1
answer
296
views
Can we inductively define Wadge-well-foundedness?
For a topological space $X$ (which I'll identify with its underlying set of points), we define the Wadge preorder $Wadge(X)$: elements of the preorder are subsets of $X$, and the ordering is given by $...
- 21.2k
6
votes
2
answers
303
views
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances?
Formal version of question. If $X$ is a set, let $[X]...
- 48.9k
4
votes
1
answer
260
views
Generalizing the $T_0$-axiom
The starting point of this question is a slight reformulation of the $T_0$ separation axiom: A topological space $(X,\tau)$ is $T_0$ if for all $x\neq y\in X$ there is a set $U\in \tau$ such that $$\{...
- 48.9k
4
votes
1
answer
224
views
Bounded growth of functions vs bounded growth of functions on countable sets
I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $...
- 5,529
4
votes
1
answer
221
views
Embedding ordinals with the order topology into connected $T_2$-spaces
Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...
- 48.9k
6
votes
1
answer
342
views
Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?
Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...
- 41.8k
9
votes
1
answer
226
views
Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group?
For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\...
- 41.8k
5
votes
1
answer
287
views
Is each compactification of $\mathbb N$ soft?
Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
- 41.8k
6
votes
2
answers
482
views
Complete atomless Boolean algebras with abelian automorphism group
Is there any example of a complete atomless Boolean algebra with a non-trivial abelian automorphism group?
This is equivalent, by Stone duality, to asking for an extremally disconnected compact ...
- 3,115
5
votes
1
answer
241
views
On filters possessing a countable network
Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$
A family $\mathcal N$ of subsets of $\omega$ is called a network ...
- 41.8k
8
votes
2
answers
1k
views
When does an "$\mathbb{R}$-generated" space have a short description?
The following is a more focused version of the original question; see the edit history if interested. In the original version of the question, five other variants of the "simplicity" ...
- 21.2k
8
votes
0
answers
240
views
Universally meager spaces and large cardinals
Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
- 18.6k
5
votes
1
answer
206
views
"König's theorem" for $T_2$-spaces?
For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\...
- 48.9k
3
votes
1
answer
112
views
Connected spaces where every dense set is large
Let $\kappa >\aleph_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every dense set $D\subseteq X$ we have $|D|=|X|$?
- 48.9k
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
9
votes
1
answer
599
views
On the Large Cardinal Strength of Normal Moore Space Conjecture
In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces:
Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable.
Then ...
3
votes
1
answer
119
views
Nice representation of open sets in $\sigma$-algebras in certain circumstances
Let $(X,\tau)$ be a topological space. For a given topological base $\mathcal{E}$ for $\tau$, let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Assume ...
- 4,058
4
votes
1
answer
121
views
Nice arrangement of open sets in $\sigma$-algebras
Let $X$ be a topological space and $\mathcal{E}$ be a topological base for $X$. Let us denote Bor$(\mathcal{E})$, by the smallest $\sigma$-algebra containing $\mathcal{E}$.
Q. Let $O$ be an open ...
- 4,058
3
votes
1
answer
285
views
Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$
Definitions: Let $X$ be a Polish space (separable completely metrizable topological space).
A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...
- 623
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
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
2
votes
2
answers
122
views
Thick refinements of covers
Let $(X,\tau)$ be a topological space, and let ${\cal U}$ be an open cover. We say that ${\cal U}$ is thick if for all $x\in X$ we have $$|\{V\in {\cal U}: x\in V\}| = |X|.$$
Is there a Hausdorff ...
- 48.9k
3
votes
0
answers
88
views
Which spaces are still Lindelöf after forcing with a Suslin tree?
Let $T$ be a Suslin tree and $f:T\to Y$ be continuous. ($T$ is endowed with the order topology.) Assume that the image of $T$ is contained in a Lindelöf subset of $Y$. Then, force with $T$. Which ...
- 1,855
5
votes
1
answer
2k
views
Nice partition of $\mathbb{R}$ into uncountably many uncountable sets
A recent issue of American Math. Monthly has a paper that partitions
$\mathbb{R}$ into an arbitrary finite number of uncountable sets such
that every real number is a condensation point of all the ...
- 50.8k
6
votes
6
answers
487
views
If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$?
If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we necessarily have $|X| =|\tau|$?
- 48.9k
3
votes
1
answer
123
views
Approximation on separable topological space with size $\mathfrak{c}$
Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function.
Q. Is it possible to find a sequence of ...
- 4,058
4
votes
0
answers
143
views
A point concerning Fremlin's example on Borel sets in non-separable Banach spaces
Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...
- 4,058
1
vote
0
answers
280
views
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
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
1
vote
1
answer
217
views
Connected Hausdorff spaces with different cardinalities of open sets
Given an infinite cardinal $\kappa$, is there a connected Hausdorff space $(X,\tau)$ with $|X|=\kappa$, and for every infinite cardinal $\lambda \leq \kappa$ there is an open set $U\in \tau$ with $|U| ...
- 48.9k
8
votes
2
answers
205
views
Spaces without maximal homogeneous subspaces
A homogeneous space $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As a previous question implies, the union of an ...
- 48.9k
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
5
votes
0
answers
140
views
How big is the smallest nontrivial partition of the unit interval into closed disjoint closed sets? [duplicate]
Consider how we might partition the unit interval in the reals into
disjoint closed sets
$$[0,1]=\bigsqcup_i C_i.$$
Of course, we could partition the unit interval into singletons, which would make ...
- 236k
2
votes
3
answers
235
views
Example of an $\omega_1$ decreasing chain of dense semicontinua?
In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows:
We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...
- 1,955
5
votes
1
answer
227
views
How many disjoint compact sets are needed to form a connected compactum?
Let's assume all spaces are metrizable. For each connected compact space $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into non-empty compact sets, excluding the trivial partition $\{X\}...
- 3,317
14
votes
1
answer
604
views
Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible
Let $X$ be a co-finite topological space. If $|X| \ge 2^{\aleph_0}=\mathfrak c$, then $X$ is contractible (https://en.wikipedia.org/wiki/Contractible_space) . Indeed, there is a bijection $f: X \times ...
user111524
14
votes
2
answers
841
views
Proper topological spaces
Recall that a topological space is ccc, or has the countable chain condition, if every family of pairwise disjoint open sets is countable.
But equivalently, we can say that the forcing defined with ...
- 39.7k
3
votes
1
answer
165
views
Nonmetrizable Corson compacta with ccc
It is known that under $MA+ \neg CH$, every Corson compact space with the countable chain condition (ccc) is merizable. It is also known that, under $CH$, there exist nonmetrizable Corson compact ...
- 419
8
votes
2
answers
289
views
Does $\aleph_0$-density of regular open algebra entail existence of countable basis?
Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as Boolean algebra) be $\aleph_0$.
Does $X$ ...
- 1,112
4
votes
0
answers
195
views
A kind of 0-1 law?
Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire,
if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
- 2,281
3
votes
1
answer
436
views
Stone topological Boolean algebras
I am looking for an initial reference for a theorem which is known, namely:
Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
- 774
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 ...
- 41.8k