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 $Z$ ...