All Questions
Tagged with set-theory gn.general-topology
433 questions
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
4
votes
0
answers
115
views
Dimension properties of some concrete hereditarily disconnected subspaces of the Hilbert cube
This question was motivated by this MO-question asking about the example of a hereditarily disconnected metrizable separable space, which is not the union of countably many totally disconnected ...
- 41.9k
19
votes
0
answers
563
views
What algebraic properties are preserved by $\mathbb{N}\leadsto\beta\mathbb{N}$?
Given a binary operation $\star$ on $\mathbb{N}$, we can naturally extend $\star$ to a semicontinuous operation $\widehat{\star}$ on the set $\beta\mathbb{N}$ of ultrafilters on $\mathbb{N}$ as ...
- 20.5k
4
votes
0
answers
333
views
Is this result of Hajnal and Juhász correct?
I am having some trouble with the following result presented here:
Obviously I'm missing something, but I think from that result it could be shown that if $X$ is an infinite topological space, then $...
- 674
5
votes
2
answers
188
views
$|\mathsf{RO}(X)|$ vs. $2^{d(X)}$ for $T_3$ spaces
Let $\mathsf{RO}(X)$ stand for the collection of regular open subsets of a topological space $X$ and let $d(X)$ be its density. It is well-known (see Theorem~3.3 of Hodel's chapter in the Handbook) ...
- 674
1
vote
0
answers
84
views
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
3
votes
0
answers
79
views
Is $\mathfrak q_0$ equal to the smallest cardinality of a second-countable $T_1$-space which is not a $Q$-space?
A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and ...
- 41.9k
4
votes
1
answer
279
views
Product topology from two premetric spaces induced by sum of premetrics?
For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$.
Do ...
- 463
4
votes
1
answer
194
views
Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?
Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
- 41.9k
7
votes
1
answer
379
views
What is an example of a meager space X such that X is concentrated on countable dense set?
A topological space $X$ is concentrated on a set $D$ iff for any open set $G$ if $D\subseteq G$, then $X\setminus G$ is countable.
What is an example of a separable metrizable (uncountable) meager (...
- 1,101
2
votes
2
answers
252
views
When is the Minkowski sum of weighted compact sets $w_1 B_1 + w_2 B_2 + \ldots$ (with $w \in L^1$) closed?
Let $B_1,B_2,\ldots,$ be compact subsets of $\mathbb R^d$ and $w_1,w_2,\ldots$ be nonnegative numbers summing to $1$. Consider the set
$$
A := w_1 B_1 + w_2 B_2 + \ldots = \left\{\sum_{n=1}^\infty w_n ...
- 6,853
3
votes
1
answer
192
views
Co-analytic $Q$-sets
A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
- 2,286
2
votes
1
answer
154
views
Images of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$
My question is:
Is every Polish space image of a closed and continuous mapping with domain $\Bbb{N}^\Bbb{N}$?
Where a Polish space is a separable and completely metrizable space and where $\Bbb{N}^\...
- 2,286
4
votes
1
answer
104
views
Are there such a complete metric space X of weight k (w(X)=k) and ....?
Are there such a complete metric space $X$ of weight $k<\mathfrak{c}$ ($w(X)=k$) and a family $\{F_{\alpha}: \alpha<k\}$ of closed subsets of $X$ that $k<|X\setminus \bigcup F_{\alpha}|<\...
- 1,101
3
votes
0
answers
77
views
What is the name of the (possibly well-known) class of $\pi$-monolithic compact spaces?
A compact space $X$ is called ${\it \pi-monolithic}$ if whenever a surjective continuous mapping $f:X\rightarrow K$ where $K$ is a compact metric space there exists a compact metric space $T\subseteq ...
- 1,101
2
votes
1
answer
265
views
Special version of $\Delta$-system Lemma for singular cardinals
In his article "Remarks on cardinal invariants in topology" (you can get the paper here: Where can I find the following S. Shelah's paper?), Saharon Shelah states the following claim:
(...
- 674
3
votes
2
answers
2k
views
Can every real function be approximated with a Riemann-integrable one with any precision required?
Is there some proof that Riemann-integrable functions are dense in the space of all real functions?
In a sense that for every real function $f$ and number $\varepsilon>0$, there is Riemann-...
- 117
6
votes
0
answers
210
views
Failure of Baire's grand theorem when the hypothesis is weakened to separable metric space
The statement of Baire grand theorem gives a characterization of Baire class 1 functions between a completely metrizable separable space (aka Polish space) and a separable metrizable space. The ...
- 2,286
4
votes
1
answer
141
views
Hedgehog of spininess $κ$ is an absolute retract?
Let $κ$ be an infinite cardinal, $S$ a set of cardinality $κ$, and let
$I = [0, 1]$ be the closed unit interval. Define an equivalence
relation $E$ on $I × S$ by $(x,α) E (y,β)$ if either $x = 0 = y$
...
- 1,101
0
votes
0
answers
94
views
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
1
vote
1
answer
149
views
Does there exist a star-Lindelöf space which is not DCCC?
A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V$ of $\mathcal U$ such that $\operatorname{St}(\bigcup\mathcal V,\mathcal ...
- 505
1
vote
2
answers
543
views
Subsets of the Cantor set
A copy of the Cantor set is a space homeomorphic to $2^{\omega}$.
Suppose that $X$ is a Hausdorff space that contains a copy $C^{\prime}$ of the Cantor set. Let $U$ be a nonempty subset open in $C^{\...
- 31
4
votes
2
answers
164
views
$|\mathsf{RO}(X)|$ vs. $|\tau_X|$ for Tychonoff spaces
Let $\tau_X$ denote the collection of open subsets of a topological space $X$ and let $\mathsf{RO}(X)$ be the subset of $\tau_X$ made up of regular open subsets. With this terminology, the inequality ...
- 674
3
votes
0
answers
122
views
A space with independent tightness
Recall that the tightness of a topological space $X$ is defined as the least cardinal $\kappa$ such that for every non-closed subset $A$ of $X$ and every point $x \in \overline{A} \setminus A$, there ...
- 4,413
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
2
votes
1
answer
167
views
Is $\mathbb R$ with cocountable topology star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 505
3
votes
1
answer
140
views
$i$-weight of a metrizable space
Recall that the $i$-weight of a Tychonoff space $X$, $iw(X)$, denotes the minimal weight of all Tychonoff spaces onto which $X$ can be condensed. A standard fact about this cardinal number is that the ...
- 674
0
votes
1
answer
157
views
Does there exist a star-Lindelöf space which is not star-$L$-Lindelöf?
A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A ...
- 505
1
vote
1
answer
117
views
Does there exist a starcompact space which is not star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 505
4
votes
1
answer
195
views
Consistency of the Hurewicz dichotomy property
Just to fix the environment, let's work in the Baire space $\omega^\omega$, the space of infinite sequences of natural numbers with the product of the discrete topology over $\omega$. We say that a ...
- 2,286
7
votes
1
answer
199
views
Scattered hereditarily separable does not imply countable in ZFC
Recall that a topological space $X$ is scattered if and only if every non-empty subset $Y$ of $X$ contains at least one point which is isolated in $Y$. Consider the statement: "Every scattered ...
- 674
2
votes
0
answers
200
views
A question about infinite product of Baire and meager spaces
Proposition 1: For any space $X$ and an infinite cardinal $\kappa$, the product $X^{\kappa}$ is either meager or a Baire space.
Does anyone have any suggestions to demonstrate Proposition 1?
I was ...
- 561
8
votes
1
answer
393
views
Where can I find the following S. Shelah's paper?
I've been trying to find the following article: "S. Shelah, Remarks on cardinal invariants in topology, General topology Appl. 7(3) (1977), 251-259". I tried to go directly to the journal ...
- 674
10
votes
0
answers
364
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
3
votes
1
answer
263
views
Give an example of a star-Menger space which is not star-$K$-Menger
A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...
- 505
3
votes
2
answers
270
views
Is the set of $\kappa$-complete ultrafilters closed in $\beta X$?
Given an arbitrary set $X$, let $\beta X$ be the set of all ultrafilters over $X$. Consider endowing $\beta X$ with a topology consisting of the following open sets:
$$
\{\mathcal{U} \in \beta X : A \...
- 1,432
5
votes
1
answer
198
views
The cardinal characteristic $\mathfrak r_{(X,f)}$ of a dynamical system
I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum.
For a compact ...
- 41.9k
3
votes
0
answers
83
views
Does there exist a regular $P$-space which is strongly star-Lindelof but not star-Menger?
A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...
- 505
5
votes
1
answer
155
views
Can the Boolean group $C_2^\omega$ be covered by less than $\mathfrak b$ nowhere dense subgroups?
Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that
$$\...
- 41.9k
8
votes
1
answer
482
views
VC dimension of standard topology on the reals
Can there be an uncountable set $S\subseteq\mathbb R$ such that for each subset $D\subseteq S$, there is an open set $U$ with $D=S\cap U$?
I'm asking merely out of curiosity, but I'll mention that ...
- 24.8k
4
votes
1
answer
259
views
Reference request: large generalized probability measures
I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size?
I'...
- 1,155
2
votes
1
answer
145
views
Uniquely selecting points from open pairwise disjoint refinements of an open cover
Let $\mathcal U$ be an open cover of some space $X$.
Let $\{\mathcal V_\alpha:\alpha<\kappa\}$ enumerate all of its pairwise-disjoint
open refinements.
When is it possible to define sets $Z_\alpha$ ...
- 1,352
18
votes
1
answer
1k
views
A topological version of the Lowenheim-Skolem number
This is a continuation of an MSE question which received a partial answer (see below).
Given a topological space $\mathcal{X}$, let $C(\mathcal{X})$ be the ring of real-valued continuous functions on $...
- 20.5k
9
votes
1
answer
410
views
On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces
In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are ...
- 3,011
9
votes
2
answers
451
views
Convergence properties in dense subsets of $\omega^*$
The space $\omega^*$, the remainder of the Cech-Stone compactification of the integers, fails to have all convergence-type properties known to me.
Sequentiality. (As a matter of fact $\omega^*$ does ...
- 4,413
6
votes
0
answers
163
views
Free sequences and the cardinality of a topological space
One way of formulating Arhangel'skii's celebrated theorem about the cardinality of Lindelof first-countable spaces is the following (due to Arhangel'skii and Shapirovskii). For every Hausdorff space $...
- 4,413
19
votes
1
answer
657
views
A large separable space of countable tightness
Is there a ZFC example of a Tychonoff space $X$ such that:
$X$ is separable.
$X$ has countable tightness (that is, a subset of $X$ is closed if and only if it contains the closure of each one of its ...
- 4,413
5
votes
0
answers
170
views
Can maximal filters of nowhere meager subsets of Cantor space be countably complete?
Let $X$ denote Cantor space. A subset $A\subseteq X$ is nowhere meager if for every non-empty open $U\subseteq X$, we have $A\cap U$ non-meager. We call $\mathcal{F}\subseteq \mathcal{P}(X)$ a maximal ...
- 369
3
votes
1
answer
179
views
A Borel perfectly everywhere dominating family of functions
Is there a Borel function $f:2^\omega\to\omega^\omega$ such that for every nonempty closed perfect set $P\subseteq 2^\omega$, $f|P$ is a dominating family of functions in $\omega^\omega$?
This is a ...
- 3,115