# Questions tagged [gn.general-topology]

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

4,124
questions

1
vote

0
answers

47
views

### Space of valuations is spectral space and what does it mean to say that conditions are closed conditions

I am reading lecture 3 of Conrad notes (link : https://math.stanford.edu/~conrad/Perfseminar/ ), in which he proves space of valuations is a spectral space. Last theorem of lecture 3.
We have a map $j:...

0
votes

0
answers

52
views

### What is compatibility?

This is rather subjective. But when we say "a measure is compatible with the topology" what do we mean exactly?
Disclaimer:
I'm not being sarcastic. I'm not being mathematically hostile. ...

2
votes

0
answers

106
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 $...

13
votes

3
answers

1k
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 $\...

2
votes

1
answer

129
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:= \...

5
votes

1
answer

111
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 ...

1
vote

1
answer

71
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 ...

4
votes

0
answers

79
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

434
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 $...

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 \...

2
votes

0
answers

103
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

187
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

184
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 $X$ ...

5
votes

1
answer

179
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 $\tau$ ...

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

154
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^{...

1
vote

1
answer

299
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.,
...

4
votes

0
answers

112
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

253
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

108
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 $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ ...

11
votes

1
answer

393
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 ...

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\...

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 $...

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

189
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 ...

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,...

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 $n\in \omega$, $\...

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 \...

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 ...

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
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 ...

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 ...

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 ...

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 ...

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 ...

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 $n$-...

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 $Z$ ...