All Questions
Tagged with gn.general-topology reference-request
325 questions
7
votes
2
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2k
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Every real-valued continuous function on a closed set of compact Hausdorff space has an extension.
I've noted, that the following fact can be proven in a few lines using $C^*$-algebra theory. I wonder if it has a simple elementary proof or not. Probably you can give me a reference.
Suppose $X$ ...
- 1,284
7
votes
1
answer
2k
views
If $S\subset\mathbb R$ is a $G_\delta$, is there a function $\mathbb R\to\mathbb R$ continuous exactly on $S$?
Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function $\...
- 483
7
votes
3
answers
310
views
Non-metric topological continua
What important results hold for non-metric continua, or where can I find a survey of such results?
There are three definitions of a continuum around: a non-empty topological space that is
(1) ...
- 504
7
votes
1
answer
389
views
References for higher descriptive set theory surveys
A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
- 3,104
7
votes
1
answer
395
views
Approximation of topological dynamical systems?
I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...
- 141
7
votes
2
answers
608
views
What is the name for a set endowed with a Lipschitz structure?
I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
- 41.9k
7
votes
1
answer
181
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Lachlan on topology for priority arguments
There is a set of notes by Lachlan from 1973 on casting priority arguments in topological language; references to these notes are few and far between, but one source refers to them as "Topology for ...
- 20.5k
7
votes
1
answer
170
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Topological rigidity of cartesian product with $\mathbb{R}$
It seems that the following is true : if $V$ and $W$ are compact differentiable manifold of the same dimension, and $\mathbb{R} \times V$ is diffeomorphic to $\mathbb{R} \times W$, then $V$ and $W$ ...
7
votes
1
answer
1k
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Reference request: norm topology vs. probabilist's weak topology on measures
Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
- 710
7
votes
1
answer
399
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Objects whose morphisms are Lipschitz maps
I recently wondered what are the spaces whose morphisms are Lipschitz maps (by which I mean: "locally Lipschitz").
The answer seems pretty clear, and proceeds like the definition of manifolds:
1) If $...
- 250
7
votes
1
answer
331
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A metric characterization of Hilbert spaces
In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
- 41.9k
7
votes
1
answer
183
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Stability Question for Isotopies Between Compact Sets
Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$.
Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
- 439
7
votes
0
answers
349
views
An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
- 1,511
7
votes
0
answers
493
views
A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
- 99
7
votes
0
answers
119
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The automorphism group of the fibered cylinder
My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
- 41.9k
7
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0
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305
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Generalizing Gromov Hausdorff distance using Vietoris topology
There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for ...
- 1,630
7
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0
answers
266
views
Remote points in $\beta X$
It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
- 79
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 41.9k
6
votes
1
answer
353
views
A strong Borel selection theorem for equivalence relations
In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16):
Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
- 365
6
votes
2
answers
309
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Reference: If $X$ is metrizable, then $X$ is realcompact iff $|X|$ is non-measurable
Note: What I call a measurable cardinal seems to be non-standard among set theorists, and should be called a $\sigma$-measurable cardinal.
I know that a discrete space is realcompact iff its non-...
- 1,211
6
votes
1
answer
678
views
Is it possible to define a closure operator in terms of partial ordering?
For boolean algebra, let's take Roman Sikorski's Boolean Algebras as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that ...
- 327
6
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3
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582
views
profinite spaces coming from profinite groups
This is probably well-known:
Does every nonempty profinite space occur as the underlying space of a profinite group? If not, which conditions have to be imposed?
- Is every profinite group ...
- 63.1k
6
votes
1
answer
246
views
Is the projectivization of a topological vector space Tychonoff?
Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\...
- 5,529
6
votes
1
answer
222
views
Example of a $G$-sphere that is not a $G$-representation sphere
Let $G$ be a finite group with the discrete topology. To set terminology:
a $G$-sphere is a sphere equipped with a continuous $G$-action
a $G$-representation sphere is a $G$-sphere obtained from an ...
- 6,792
6
votes
1
answer
205
views
Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$
It is a well known fact that if $\mathcal{F}$ is a non-principal ultrafilter on $\omega$, then the set $\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$ (conflating binary strings with subsets of $\...
- 12.5k
6
votes
1
answer
186
views
Reference request: A collection of topologies on $\mathbb{N}$ formed via series
First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...
- 6,546
6
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1
answer
339
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Factorization of a certain map through a CW-complex
Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...
- 423
6
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1
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478
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Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets?
Is the following statement true, and if it is, does someone have a reference?
Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, =...
- 32.5k
6
votes
1
answer
815
views
When is a Topological pushout also a Smooth pushout?
I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:
Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram ...
- 732
6
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0
answers
255
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 ...
- 2,286
6
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0
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309
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,689
6
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0
answers
131
views
A theorem by R.L. Moore
The following result is due to R.L. Moore.
Let $K\subseteq\mathbb C$ be compact. Suppose that
$K$ is connected,
and that $\mathbb C\setminus K$ is connected.
Then $\partial K$ is connected.
Does ...
- 687
6
votes
1
answer
300
views
Proof of Denjoy-Riesz Theorem and Moore's Generalization?
The Denjoy-Riesz Theorem states that any compact zero-dimensional subset of the plane can be covered by an arc, i.e. an embedded image of $[0,1]$. Sometimes it's stated just for covering a Cantor Set,...
- 439
6
votes
0
answers
170
views
Whitney stratification for proper morphisms
Let $f: X \to \Delta$ be a flat, projective morphism, smooth over the punctured disc $\Delta^*:=\Delta \backslash \{0\}$ and central fiber $f^{-1}(0)$ is a reduced, simple normal crossings divisor. ...
- 1,593
6
votes
0
answers
132
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Generalization of pseudogroups
Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup
One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
- 181
6
votes
0
answers
563
views
Grothendieck letter to Jun-Ichi Yamashita on tame topology
I am looking for Grothendieck writings on tame topology:
a manuscript on tame topology mentioned by Scharlau; a letter to Jun-Ichi Yamashita; a letter to Z.Mebkhout.
I am also interested in ...
- 99
5
votes
2
answers
1k
views
An example of an open discontinuous function
Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the ...
- 1,897
5
votes
1
answer
206
views
If a subspace $F$ is contained in a subspace $G$, and $H$ is close to $G$, can we choose a subspace of $H$ close to $F$?
Let $E$ be a Banach space. Recall that the collection of all closed linear subspaces of $E$ can be turned into a metric space in a number of ways. In particular, consider the notion of a gap: if $G$ ...
- 5,529
5
votes
2
answers
502
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A few standard results (on metrizability and relative separation strength) without Choice?
I've been going back over some results from Munkres's Topology, and I'm curious about some things. (I originally posted this on M.SE, but I think it is probably a better fit here.)
I know that Choice ...
- 503
5
votes
1
answer
654
views
Fréchet L-Spaces
According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...
- 1,093
5
votes
1
answer
511
views
Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
- 1,167
5
votes
1
answer
494
views
When is a $*$-homomorphism between multiplier algebras strictly continuous?
(This question was posted on MSE here but didn't get any answers.)
The strict topology on the multiplier algebra M(A) of a C*-algebra A is that generated by the seminorms
$$ x\mapsto \|ax\|\quad x\...
- 1,336
5
votes
1
answer
247
views
Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?
If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
- 213
5
votes
2
answers
448
views
Space of curves
I am reading Burago, Burago & Ivanov's book where they distinguish the notion of a curve and a path in the following way:
a path in a topological space $X$ is simply a (continuous) map from a ...
- 5,529
5
votes
1
answer
216
views
Continuity of taking collapse maps
Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
- 666
5
votes
1
answer
247
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Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)
Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...
- 666
5
votes
0
answers
113
views
Stronger form of countable dense homogeneity
I am completing my undergrad thesis about topological properties of some subspaces of the real numbers, and CDH spaces are one of the topics I´ve covered (I know almost nothing about it, I only prove ...
- 10.6k
5
votes
0
answers
481
views
Open convex hull of a closed set
Let $X$ be a closed set in a Euclidean space of finite dimension and suppose that its convex hull $H$ is open. I can prove that, in this case, $H$ is a Cartesian product of a line with an open convex ...
- 18.6k
5
votes
0
answers
330
views
The second dual of $C(X)$ with the compact-open topology
Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
- 233
5
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0
answers
265
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Quotienting disk inside sphere result in sphere
Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let
$q: D^k \to D^r$ be a map and $r \leq k$. Let
$$W = S^k \sqcup D^r/\sim$$
where $S^...
- 2,023