Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

Filter by
Sorted by
Tagged with
2 votes
0 answers
77 views

Which cardinal $\kappa\geq \omega_1$ is critical for the following property...?

Which cardinal $\kappa\geq \omega_1$ is critical for the following property: Let $X\subset \mathbb R$ and $\kappa>|X|\geq \omega_1$. Then there is an uncountable family $\{X_{\alpha}\}$ such that $...
7 votes
2 answers
509 views

Is there a universal property characterizing the category of compact Hausdorff spaces?

This is in some sense a follow up to the question asked here Properties of the category of compact Hausdorff spaces To clarify: The category $\text{Prof}$ of profinite sets sits inside the category $\...
  • 1,663
2 votes
1 answer
122 views

Open sets in the space of signed measures equipped with the Kantorovich–Rubinshtein norm

Let $X$ be a compact metric space and $\mathcal{M}(X)$ be the space of variational-bounded, signed Borel measures equipped with the Kantorovich–Rubinshtein norm, cf. [Section 8.3, 1]: $$||\mu||_0:= \...
  • 135
5 votes
1 answer
104 views

Scott topology: Suprema of sequences are topological limits

I read that, with the Scott topology, suprema of sequences are topological limits (See page 1 of this article). Let $(X, \le)$ be a DCPO, and $D$ be a directed subset of $X$. I can easily see that the ...
  • 464
1 vote
1 answer
70 views

abstract description of the topology on a real vector space defined by the algebraically open sets

Let $V$ be a real vector space. Given a subset $A \subseteq V$, say that a point $x \in A$ lies in the algebraic interior of $A$ if every affine line $\ell$ that passes through $x$ has the property ...
  • 241
4 votes
0 answers
76 views

A generalized Hausdorff dimension in form of a Lower semi continuous function

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
11 votes
2 answers
416 views

Existence of an open convex set

Let $T$ be a normed vector space, $K\subseteq T$ compact and convex and $O\subseteq K$ convex and open in $K$ (i.e. open w.r.t. the subspace topology of $K$ inherited by $T$). Can we find an open set $...
  • 135
0 votes
0 answers
61 views

G separable group, $\aleph_0 \leq \tau$. What is $l(X)$ and $\omega l(X) (\leq \tau)$? where $X \subseteq G$. And what is $\chi (G)$ (cardinal)?

Happy Chinese new year! I was reading (and translating) a Russian article "On the topological groups close to being Lindelöf". Where it is assumed G is a separable group and $\tau \geq \...
1 vote
0 answers
98 views

Is there a Lusin space $X$ such that ...?

Is there a Lusin space (in the sense Kunen) $X$ such that $X$ is Tychonoff; $X$ is a $\gamma$-space ? Note that if $X$ is metrizable and a $\gamma$-space then it is not Lusin. In mathematics, a ...
3 votes
2 answers
185 views

Uniformly continuous homotopy equivalence

Suppose $M$ and $N$ are complete metric spaces and $f, g: M \to N$ are uniformly continuous maps between them with common modulus of continuity $m$. Further suppose $f$ and $g$ are homotopy equivalent....
4 votes
0 answers
183 views

Almost compact sets

Update: Q1 is answered in the comments. I think that the usual arguments show that every relatively almost compact set in a space is closed in the space. Original question: A set $K$ in a space $X$ ...
  • 2,624
5 votes
1 answer
171 views

Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?

Recall that $\mathfrak{p}=\min\{|F|: F$ is a subfamily of $[\omega]^{\omega}$ with the sfip which has no infinite pseudo-intersection $\}$. The cardinal $\mathfrak{q}_0$ defined as the smallest ...
14 votes
1 answer
207 views

Is there a countably infinite closed interval in the lattice of topologies?

Is there an interval of the form $[\sigma,\tau]$ in the lattice of topologies on some set $X$ such that $|[\sigma,\tau]| = \aleph_0$? In other words, do there exist two topologies $\sigma$ and $\tau$ ...
  • 15.7k
14 votes
1 answer
478 views

How “disconnected” can a continuum be?

A continuum is a compact connected metrizable topological space. Given a cardinal $\kappa$, a topological space $X$ is called $\kappa$-connected if it is not possible to write $X$ as the disjoint ...
3 votes
0 answers
153 views

What is the name of the class of topological spaces with the following property ....?

What is the name of the class of topological spaces with the following property $P$ ? $X\in P$ iff for any open set $W$ in $X$ and any point $x\in \overline{W}\setminus W$ there is an open set $V$ ...
3 votes
1 answer
82 views

Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?

A topological space $X$ is called a $\sigma$-space if every $F_{\sigma}$-subset of $X$ is $G_{\delta}$. A topological space $X$ is called a $Q$-space if any subset of $X$ is $F_{\sigma}$. Definition. ...
3 votes
0 answers
130 views

Extending continuous maps from spheres to Euclidean spaces [migrated]

Fix $d\in\mathbb{N}$. Consider the following sets as topological spaces with the subspace topology from $\mathbb{R}^{d+1}$. $$S^d = \{ (x_0,\ldots,x_d)\in\mathbb{R}^{d+1}\mid \sum x_i^2 = 1\}$$ $$ D^{...
  • 713
1 vote
1 answer
298 views

A question about realcompact spaces

Let $X$ be completely regular space, $\beta X$ be Stone-Čech compactification of $X$, and $\upsilon X$ be Hewitt realcompactification of $X$. Then $X\subset \upsilon X\subset \beta X$. If the ...
1 vote
0 answers
162 views

the Brouwer fixed point theorem for maps rather than spaces

Is there a version for the Brouwer fixed point theorem for maps rather than spaces ? In other words, for a family of endomorphisms, can the fixed point be chosen continuously, under some assumptions ? ...
6 votes
1 answer
143 views

Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?

Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the Bochner integral. Then we have Theorem 1.40 in Rudin's Real and Complex Analysis, i.e., ...
  • 527
4 votes
0 answers
110 views

An uncountable Baire γ-space without an isolated point exists?

An open cover $U$ of a space $X$ is: • an $\omega$-cover if $X$ does not belong to $U$ and every finite subset of $X$ is contained in a member of $U$. • a $\gamma$-cover if it is infinite and each $x\...
12 votes
3 answers
750 views

Fixed point theorem for the uncountable power of an interval

Does the Brouwer fixed point theorem holds for the uncountable power $[0,1]^\kappa$ of the interval, $\kappa\geq\aleph_1$ ? That is, does every continuous endomorphism $[0,1]^\kappa\to [0,1]^\kappa$ ...
6 votes
1 answer
246 views

When is a contractible space a retract of the Hilbert cube or $\Bbb R^\omega$?

Which contractible spaces appear as retracts of the Hilbert cube or of $\Bbb R^\omega$ ? One wants to think that a sufficiently “nice” contractible space is necessarily a retract of the Hilbert cube ...
1 vote
1 answer
107 views

When are fixed point sets in $T_1$ spaces always closed?

Let $X$ be a topological space, and say that $X$ satisfies the closed fixed point set property if every continuous self-map $f:X\to X$ has fixed point set $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ ...
  • 2,211
11 votes
1 answer
392 views

A topological tree is weakly contractible

Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
  • 163
9 votes
0 answers
132 views

Irreducible subcontinuum without Zorn's lemma

In continuum theory we frequently use the fact that two points in a continuum are contained in an irreducible subcontinuum. A continuum $X$ is a compact connected metric space. A subcontinuum $K\...
  • 2,854
4 votes
1 answer
223 views

"Weird-open" maps in topology

Given topological spaces $X$ and $Y$, we define an open map from $X$ to $Y$ to be a map of sets $f\colon X\to Y$ satisfying the following condition: For each $U\in\mathcal{P}(X)$, if $U$ is open in $...
  • 1,143
4 votes
2 answers
157 views

Which topological spaces have a standard Borel $\sigma$-algebra?

Call a topological space $X$ standard Borel if $X$ is standard Borel as a measurable space (equipped with its Borel $\sigma$-algebra), i.e. if there is a Borel isomorphism between $X$ and a Polish ...
4 votes
0 answers
153 views

Brouwer fixed point theorem for non-Hausdorff spaces

Can the Brouwer fixed point theorem be formulated for non-Hausdorff spaces ? More particularly, is there a formulation of the Brouwer fixed point theorem which covers both the standard case of ...
5 votes
1 answer
187 views

Does the Rieger-Nishimura lattice over a subset of $\mathbb{R}^k$ stabilize?

Notation: If $U,V$ are open subsets of a topological space $X$, let us write $U\Rrightarrow V$ for the Heyting operation: the largest open subset $W$ of $X$ such that $U\cap W \subseteq V$ (i.e., the ...
  • 24.5k
3 votes
0 answers
55 views

Closure of the inverse image under the projection map

Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
  • 51
2 votes
0 answers
56 views

a connected geometrically unibranch algebraic stack of finite type over a field is irreducible

Let $f:X\to \mathfrak{X}$ be a smooth presentation of geometrically unibranch connected algebraic stack by a scheme, which is geometrically unibranch since being geom. unibranch is local in smooth ...
3 votes
1 answer
101 views

On the Menger property and the Alexandroff duplicate

Recall that a space $X$ is Menger if for each sequence $(\mathcal{U}_n)_{n\in\omega}$ of open covers of $X$, there is a sequence $(\mathcal{V}_n)_{n\in\omega}$ such that, for each $n\in \omega$, $\...
  • 146
15 votes
1 answer
417 views

Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?

Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
  • 24.5k
1 vote
1 answer
77 views

Is the class of rc-spaces closed under products?

Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r:...
0 votes
0 answers
119 views

Cyclic group action and finite invariant set

Let $(X, d)$ be a compact metric space and $G$ a discrete group acting on $X$ such that, for each $g\in G$, the mapping $x\mapsto g\cdot x$ defines a homeomorphism on $X$ Is it true that the ...
3 votes
0 answers
87 views

"Practical" references on mapping spaces as infinite-dimensional manifolds

I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
  • 475
5 votes
0 answers
128 views

Every Polish space is the image of the Baire space by a continuous and closed map, reference

The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501) Every Polish space (i.e. every separable ...
  • 1,423
1 vote
1 answer
41 views

Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?

Let $(E, d)$ be a Polish space and $\mathcal C_b(E)$ the space of all real-valued bounded continuous functions on $E$. Let $p \in [1, \infty)$ and $\mathcal P_p (E)$ the space of all Borel probability ...
  • 527
1 vote
1 answer
54 views

Conditions that ensure the metric topology of $E$ coincides with the initial topology induced by a collection of real-valued functions on $E$

Let $(E, d)$ be a metric space and $\mathcal F$ a collection of real-valued functions on $E$. Assume that for all $x,x_n \in E$ with $n\in \mathbb N$, $$ x_n \to x \iff [f(x_n) \to f(x) \quad \forall ...
  • 363
9 votes
1 answer
148 views

Is there a connected Hausdorff anticompact space that is countably infinite?

Cross-posted from MSE. Following Bankston - The total negation of a topological property, a topological space is called anticompact if all its compact subsets are finite. The linked MSE post above ...
  • 193
1 vote
1 answer
77 views

A question about a realcompact space and upper semicontinuous function

Nancy Dykes says in the proof of Theorem 3.4 in her article Generalizations of realcompact spaces that by a result of John Mack, if for every $p\in \beta X\setminus X$ there exists a nonnegative upper ...
2 votes
0 answers
132 views

Concrete description of “DeMorganian” open sets

Let me begin with a few definitions. My question will be basically how to simplify them to something more manageable. The motivation for these definitions is given at the end. Let $X$ be a ...
  • 24.5k
4 votes
0 answers
206 views

Does Tychonov's theorem directly imply Zorn's lemma?

This question was formerly posted on MSE https://math.stackexchange.com/questions/4578923/ without getting an answer. I know that Tychonov's theorem, Zorn's lemma, the axiom of choice, the well-...
4 votes
1 answer
96 views

For which $X$ is $X\times I$ collectionwise normal?

Many normality-type properties can be characterised in terms of products with the unit interval $I=[0,1]$. For instance, if $X$ is a Hausdorff space, then; $X$ is normal and countably paracompact if ...
  • 3,869
6 votes
0 answers
60 views

Classification of contractible open n-manifolds which embed in a compact n-manifold

Does there exist a classification of contractible open $n$-manifolds ($n\geq 3$) which embed in a compact $n$-manifold? More general, does there exist a classification of contractible open $n$-...
  • 1,405
2 votes
1 answer
61 views

Why are the selection principle $S_\text{fin}(\Lambda, \Omega)$ and $S_\text{fin}(\mathcal{O},\Lambda)$ impossible for nontrivial spaces?

Recall that an open cover $\mathcal{U}$ of $X$ is a $\gamma$-cover if it is infinite and each $x\in X$ belongs to all but finitely many elements of $\mathcal{U}$ and an open open cover $\mathcal{V}$ ...
5 votes
0 answers
136 views

Does "achieving more GH-distances than some compact space" imply compactness?

Previously asked and bountied at MSE: For complete metric spaces $X,Y$, write $X\trianglelefteq Y$ iff for every complete metric space $Z$ such that the Gromov-Hausdorff distance between $X$ and $Z$ ...
23 votes
2 answers
585 views

"All retracts are closed" and "all compacts are closed"

I want to follow the discussion from here concerning about the strength of the separation "all retract subspaces are closed". (A retract subspace of a topological space $X$ is a subspace $A$ ...
5 votes
1 answer
165 views

How many pairwise non-homeomorphic non-empty closed subsets of the Cantor set are there? [duplicate]

My question is more or less related to basic set theory. But I don't know even that. Apologies if I added the wrong tags. Motivation: How many non-compact (planar) surfaces are there upto ...
  • 761

1
2 3 4 5
83