Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,379
questions
4
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Stone–Čech compactification and an ultrafilter of regular closed sets
$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$A subset $A$ of a topological space $X$ is called regular closed if $A=\cl
_{X}\int_{X}A$.
The family of all regular closed sets of a ...
- 1,109
3
votes
1
answer
113
views
Spectrum of continuous functions as a semigroup
Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
18
votes
0
answers
1k
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Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
- 569
4
votes
1
answer
132
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Is the set of clopen subsets Borel in the Effros Borel space?
Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
- 2,971
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0
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48
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A generalization of relative interior?
In an infinite-dimension space, the relative interior of a non-empty convex set may be empty. I was wondering whether there is a concept (as a generalization of relative interior) with the following ...
- 159
3
votes
0
answers
61
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Smooth Hamiltonian diffeomorphisms form a Baire space
Let $S$ be a closed surface equipped with an area form $\omega$. In Corollary 1.2 of this paper, Asaoka and Irie demonstrated that Hamiltonian diffeomorphisms which have a dense set of periodic points ...
- 219
0
votes
1
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88
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A question about the Stone-Čech compactification and ultrafilter
Let $X$ be a Tychonoff space and let $\beta X$ is the Stone-Čech
compactification of $X$. Assume $f:X\longrightarrow \mathbb{R}$ is a bounded function. Then there exists a function $f^{\beta }:\beta X\...
- 1,109
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votes
0
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40
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T. Isiwata's "T. Isiwata. d-, d*-maps and cb*-spaces."
I need T. Isiwata's article T. Isiwata. d-, d-maps and cb-spaces. Bull. Tokyo. Gakugei Univ. Ser. IV, 29, 1977.
Does anyone have it?
https://mathscinet.ams.org/mathscinet/article?mr=0454902
https://u-...
- 1,109
5
votes
1
answer
176
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Topological property of convergent sequences being eventually constant
Is there a name in the literature for the topological property that all convergent sequences are eventually constant?
This property seems to occur with some frequency and it would be nice to have a ...
- 311
4
votes
1
answer
192
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A problem on Demailly's proof of finiteness theorem of elliptic differential operator
I am reading Demailly's notes on pseudodifferential operators on manifolds. And I cannot understand a statement he had made when he tried to prove that the image of an elliptic differential operator ...
0
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0
answers
61
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T. Hanaoka's "Note on c-realcompact spaces and mappings"
I need T. Hanaoka's article Note on c-realcompact spaces and mappings, Memoirs of the Osaka Kyoiku Univ., Ser. Ill, 26 (1977), 55-58.
Can anyone find it for me?
http://ir.lib.osaka-kyoiku.ac.jp/dspace/...
- 1,109
2
votes
1
answer
271
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Global control of locally approximating polynomial in Stone-Weierstrass?
Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\...
- 411
3
votes
1
answer
328
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Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis?
Let me start with the following
Illustration: Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the ...
- 7,276
0
votes
0
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157
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Homeomorphism groups on manifolds and topological properties
Let $M$ be a compact $n$-dimensional manifold let $H(M)$ denote the homeomorphism group of $M$.
If $n=2$ then $H(M)$ enjoys nice properties such as being an ANR, is locally contractible, separable. ...
2
votes
1
answer
179
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$\sigma$-product of the Hilbert cube
Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$
("eventually&...
- 2,993
2
votes
2
answers
269
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Topological characterisations of properties of posets
Finite connected partially ordered sets are in bijective correspondence to connected finite topological spaces that satisfy T_0, see for example the Wikipedia article Finite topological space. Here ...
- 25.4k
2
votes
1
answer
197
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Parametrization of topological algebraic objects
There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is ...
- 5,275
2
votes
0
answers
45
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$\sigma$-compactness of probability measures under a refined topology
Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
- 195
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0
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260
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$T_1$ paratopological group having a dense commutative subgroup is commutative
I'm learning about topological groups from Arhangelskii and Tkachenko "Topological groups and related structures" and this is one of the exercises there.
A paratopological group is a group ...
- 635
8
votes
2
answers
468
views
Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
- 6,764
4
votes
2
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143
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Does there exist a non-hemicompact regular space for which the 2nd player in the $K$-Rothberger game has a winning Markov strategy?
Assume spaces are regular.
A space is $\sigma$-compact if and only if the second player in the Menger game has a winning Markov strategy (relying on only the most recent move of the opponent and the ...
- 933
6
votes
1
answer
511
views
Does Playfair imply Proclus?
I am interested in the interplay between the Playfair and Proclus Axioms in linear spaces.
By a linear space I understand a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of ...
- 40.2k
3
votes
2
answers
176
views
Maximal contractible-ish Hausdorff surfaces
For the duration of this question, let a "surface" be any connected Hausdorff topological space that is locally homeomorphic to R2. Note that we make no assumption about a countable base to ...
- 2,468
1
vote
0
answers
107
views
Refinement of an open cover for a simply connected compact subset
Let $U$ denote a simply connected, open subset of the plane, and let $K$ be a simply connected, compact subset of $U$. Can we always find a finite or countable sequence of open disks $(D_n)$ such that:...
- 341
8
votes
1
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193
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Can totally inhomogeneous sets of reals coexist with determinacy?
A special case of a theorem of Brian Scott (from On the existence of totally inhomogeneous spaces) is that there is a size-continuum set $S\subset\mathbb{R}$ such that if $x,y\in S$ are distinct then $...
- 19k
4
votes
1
answer
176
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Profinite groups with isomorphic proper, dense subgroups are isomorphic
I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
- 253
4
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0
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94
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Is the range of a probability-valued random variable with the variation topology (almost) separable?
Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
- 12.3k
6
votes
0
answers
186
views
Making the analogy of finiteness and compactness precise
If one asks about the intution behind compact topological spaces, most often one will hear the mantra
“Compactness of a topological space is a generalisation of the finiteness of a set.”
For example,...
- 1,093
1
vote
0
answers
49
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Is a “well-behaved” closed subbasis for the topology generated by a closure operator a closed basis for the closure operator itself?
Let $\Omega$ be a set, $\mathcal{c}: \mathcal{P}(\Omega) \rightarrow \mathcal{P}(\Omega)$ be a closure operator (i.e., $\mathcal{c}$ satisfies $X \subseteq \mathcal{c}(X)$ and $\mathcal{c}(\mathcal{c}(...
- 1,146
1
vote
0
answers
21
views
Weakening compacity hypothesis in multifunctions intersection
Let $X,Y$ be metric spaces, $x^*\in X$
We define two multifunctions $F_1:X\rightrightarrows Y$,$F_2:X\rightrightarrows Y$.
We recall the upper-semi-continuity in Berge's sense :
A multifunction $F:X\...
- 141
1
vote
1
answer
253
views
Does global boundedness ruin Stone-Weierstrass denseness?
Let $X$ be any topological space and denote by $\tau_X$ the topology on $C_b(X;\mathbb{R})$ that is induced by the family of seminorms $(\|\cdot\|_\psi\mid\psi\in B_0(X))$ with $\|f\|_\psi:=\sup_{x\in ...
- 411
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votes
1
answer
171
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Is the unit ball of $B(H)$ a Baire space (with the SOT)?
Let $H$ be a Hilbert space, and let $B(H)$ be the set of bounded linear operators $t \colon H \to H$. Recall that we say $t_i \to t$ in the strong operator topology if $t_i \xi \to t \xi$ for every $\...
- 713
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votes
0
answers
74
views
Completeness of a normed space
We consider the set $\mathcal{PC}([-r,0],X)$
$$\mathcal{PC}([-r,0],X):=\{\varphi:[-r, 0] \rightarrow X: \varphi \text{ is continuous everywhere except
for a finite number
of points } t_* \text{ ...
- 31
2
votes
0
answers
88
views
Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?
Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...
- 830
17
votes
0
answers
936
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"Next steps" after TQFT?
(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.)
Recently, I've been ...
- 357
5
votes
1
answer
108
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Is there an $\varepsilon$-space which is not $k$-Lindelöf?
Crossposted from https://math.stackexchange.com/questions/4717613
An $\omega$-cover $\mathscr U$ of a space $X$ is a collection of open sets so that $X \not\in\mathscr U$ and every finite subset of $...
- 290
4
votes
1
answer
214
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Mysior plane is not realcompact
Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup ...
- 635
2
votes
0
answers
139
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Properties of universal fibration
I am trying to read the following paper [1] (Becker, James C.; Gottlieb, Daniel Henry
Coverings of fibrations.
Compositio Math.26(1973)) where the authors mentioned that for any fiber $F$,
there ...
- 179
2
votes
0
answers
44
views
The world of non-weak*-topologies on $\mathcal{P}(X)$
Let $X$ be a metrizable space and consider $\mathcal{P}(X)$, the set of all probability measures on $X$.
Typically, the weak*-topology is considered on $\mathcal{P}(X)$, which is a very natural ...
- 419
28
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 6,764
4
votes
0
answers
420
views
Non-triviality of map $S^{24} \longrightarrow S^{21} \longrightarrow Sp(3)$
Let $\theta$ be the generator of $\pi_{21}(Sp(3))\cong \mathbb{Z}_3$, (localized at 3).
How to show the composition
$$S^{24}\longrightarrow S^{21}\overset{\theta}\longrightarrow Sp(3)$$
is non-trivial ...
0
votes
1
answer
94
views
A question about filterbasis
K. Hardy and R. G. Wood assert that the family in line 4 is a filterbase. I couldn't show it.
- 1,109
3
votes
1
answer
138
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Do we have uniformization theorems for fractional dimensional spaces?
The Riemann mapping theorem in $\mathbb{R}^2$ is known not to generalize well in higher dimensions and is basically trivial in lower dimensions.
I’m interested in how it generalizes for fractional ...
- 2,233
5
votes
2
answers
165
views
Polish space isometric to its hyperspace
For a Polish space $(X,d)$ its hyperspace $(K(X),d_H)$ is also a Polish space. (Here $K(X)$ denotes the set of all nonempty compact subsets of $X$, and the Hausdorff metric $d_H$ is defined by $d_H(K,...
- 157
1
vote
0
answers
39
views
Discreteness of $D^{-1}D$ given that $D$ is uniformly discrete
Let $G$ be a topological group with unit element $e$.
We say that $D\subseteq G$ is discrete if for all $x\in D$ there is a unit-neighborhood $U\subseteq G$ such that $x^{-1}D\cap U=\{e\}$. We say ...
- 111
4
votes
1
answer
208
views
Being contained in a compact set
I have a sequential, hereditarily Lindelöf topological space $\mathcal{X}$, and some subset $A \subseteq \mathcal{X}$. I am interested in the following properties:
There is some compact set $B$ with $...
- 4,096
1
vote
0
answers
95
views
Problems Correction of "Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "' [closed]
Where I can find the problems correction of this book " Algebra, Topology, Differential Calculus, and Optimization Theory For Computer Science and Machine Learning "
- 11
5
votes
1
answer
128
views
Algebraic solutions of polynomial ODEs
Given a polynomial ODE in $n$-dimensions of maximal degree $d$
$$
\dot{x}_j=f_j(x)=\sum_{i_{1},\dots,i_{n}=1}^{d}a_{i_{1},\dots,i_{n}}^{j}x_{1}^{i_{1}}\dots x_{n}^{i_{n}} \quad \forall j=1,...,n
...
- 227
0
votes
0
answers
24
views
Space with hypercomponents whose restriction to a hypercomponent is not locally finite
Problem
Give an example of a topological space $(X, \mathcal{T})$ for whose hyperconnected (aka irreducible) components $\mathcal{C} \subset \mathcal{P}(X)$ it holds that $\mathcal{C}|D$ is not $(\...
- 387
4
votes
0
answers
320
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Does a contractible locally connected continuum have an fixed point property?
I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
