Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,447
questions
6
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3
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582
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Convex subsets of an open set
Let $X\subseteq \mathbb{R}^n$ be an open set and let $Y\subseteq X$ be convex. Can I always construct an open and convex set $Z$ such that $Y\subseteq Z\subseteq X$?
Edit: As Ilya Bogdanov pointed ...
- 157
0
votes
1
answer
134
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A link between continuity and 0-borelian? [closed]
Is it true that :
1/ if $f$ real continuous and $O$ an open set then $f(O)$ is a 0-borelian?
2/ if $A$ a 0-borelian set then there exists $f$ real continuous and $O$ an open set with $A=f(O)$?
$B$ ...
- 3,801
0
votes
1
answer
493
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Is the meaning of "irreducible manifold", "not reducible to other manifold"?
This is a cross post of MSE.
Q1: What does "irreducible manifold" mean (not definition)?
My understanding of "irreducible manifold" is "is not reducible (homotopic or ...
- 4,165
1
vote
0
answers
187
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Uniform convergence over compacts subsets implies existence of a uiform convergente subsequence?
Let $H$ the group of all homeomorphisms of a locally compact second countable and totally bounded metric space $X$ onto itself, under the compact-open topology ($X$ is totally bounded if every ...
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votes
0
answers
135
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Are geometric progressions closed in the $p$-adic topology?
For a prime number $p$, the $p$-adic topology on the set $\omega$ of non-negative integers is generated by the base consisting of the arithmetic progressions $x+p^n\omega:=\{x+p^ny:y\in\omega\}$ where ...
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7
votes
2
answers
226
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Continuous section of support - Is it possible to map compact sets to measures supported on them?
Let $(X,d)$ be a compact metric space and let $(\mathcal K(X),d_H)$ and $(\mathcal P(X),d_W)$ denote its space of nonempty compact subsets with Hausdorff metric $d_H$, and its space of Borel ...
- 459
5
votes
0
answers
111
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Free topological groups as smooth infinite-dimensional manifolds
It is known that the Graev free topological group $F(I)$ of the segment $I=[0,1]$ is homeomorphic to $\mathbb R^\infty=\lim\limits_{\longrightarrow}\mathbb R^n$. Is it possible to make an ...
- 4,435
11
votes
0
answers
319
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Does any real function have a Lipschitzian restriction on $D$?
Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
- 3,801
6
votes
0
answers
162
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J. F. Adams Proof of Cellular Approximation Theorem
In Ronald Brown's discussion of the proof of The Cellular Approximation Theorem in Topology and Groupoids Sec. 7.6 he writes that, "the elegant formulation of the proof is due to J. F. Adams." Does ...
- 161
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votes
1
answer
166
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Does a homeomorphism of open cones restrict to a quotient map of the bases?
Write $CX$ for the (pointed, or reduced) cone on $X$, and $C^\circ X$ for the open cone inside of it.
Let's say a cone map is a map $g:CX\to CY$ such that $g(C^\circ X) \subseteq C^\circ Y$ and $g(X) ...
- 12.5k
3
votes
1
answer
93
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$T_2$-space with a matching equalling the density number
Given a topological space $(X,\tau)$, we say that a matching is a collection of non-empty open sets that are pairwise disjoint.
Given an infinite cardinal $\kappa$, is there a $T_2$-space with $|X|\...
- 45.5k
4
votes
0
answers
88
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Pairs not at maximal distance in compact set
Does there exist a compact subset $K$ of $\mathbb{R}^2$ and a point $p$ in $\mathbb{R}^2$ with the following property?
Let $x$ and $y$ be a pair of points in $K$ such that \begin{equation} |x-y| <...
- 1,689
4
votes
1
answer
263
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Nowhere compact subsets of the plane
Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty.
Can $X$ be densely embedded into the plane?
In other words, is there a dense set $X'\subseteq ...
- 3,055
6
votes
1
answer
535
views
Extension of Baire's Theorem
Let $X$ be a topological space, $\kappa$ be a cardinal number, such that there exists a dense subset $A\subseteq X$ of cardinality $\kappa$ but there does not exist a dense subset $A'\subseteq X$ of ...
- 4,969
4
votes
2
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600
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Question about additive subgroups of the real line and the density topology
I am new studying additive subgroups of the real line, I would like to know if someone could give me an idea for the next question.
Let $m$ be the Lebesgue measure in $\mathbb{R}$. A measurable set $E\...
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votes
3
answers
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Continuous functions taking uncountably many values countably often
Let $f$ be a continuous function defined on the closed interval $[0,1]$. Clearly $f$ is bounded and attains its bounds.
Then my question is how often can $f$ take a value in its range countably many ...
- 4,792
3
votes
1
answer
141
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Two paths to the boundary with no holes in between
Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $\...
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votes
1
answer
167
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Question about almost locally ccc and the Krom space
Definition 1. A family $\mathcal{B}$ of non-empty open sets in a topological space will be called $\pi$-base (or pseudo-base) if every non-empty open set contains at least one member of $\mathcal{B}$. ...
- 551
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votes
1
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101
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If a topological space (X,T) is completely normal, and if we double the point of X, is the resulting space also completely normal? [closed]
I have a question on my proof of the following lemma, and I'd like to know if my answer is correct.
Lemma. Suppose $(X,T)$ is any completely normal topological space. Let's double the points of $X$, ...
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votes
0
answers
249
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How to compute fundamental groups of slice disk complements?
To compute the fundamental group of the complement $S^3 \setminus K$ of a knot, one usually uses the Wirtinger algorithm. Is there a similarly well-established procedure for computing the fundamental ...
- 323
1
vote
1
answer
166
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Ordering a subset of the clopens of a Stone space
Let $P$ be a countably infinite set of propositional variables and $\mathcal{L}_P$ be the propositional language generated from $P$ and the usual connectives $\wedge$, $\neg$, $\vee$. The set $\...
- 125
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votes
1
answer
171
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A map into a Hilbert space with prescribed orthogonality
Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
Does there always ...
- 5,385
2
votes
1
answer
138
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Explicit construction of a convex metric
Let $(X,d)$ be a compact, connected, locally connected, locally compact metric space.
A result of Bing and Moise (independently) states that $(X,d)$ admits a topology preserving convex metric i.e., ...
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3
votes
1
answer
165
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Reversal of open cover with topologically transitive dynamical system
Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel ...
- 4,969
0
votes
1
answer
89
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Topologically transitive dynamical system mapping space into ball
Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
- 4,969
7
votes
1
answer
539
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What is the topology on the set of field orders
Inspired by this question I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)?
For example for the function field $\...
- 10.3k
17
votes
1
answer
2k
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Best introductory texts on pointless topology
As I understand it, there are three canonical textbooks on pointless topology: the classic "Stone Spaces" by Johnstone, "Topology via Logic" by Steve Vickers, and the newer "Frames and Locales" by ...
user147820
10
votes
2
answers
351
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Source on smooth equivalence relations under continuous reducibility?
This question was asked and bountied at MSE, but received no answer.
In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
- 19.3k
0
votes
1
answer
131
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A closed subset $B$ of the Hilbert cube such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set
$\def\cube{\mathbf Q}$Let $\cube$ be the Hilbert cube. Give an example of a closed subset $B$ of $\cube$ such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set.
If anyone has any idea ...
1
vote
0
answers
116
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Stone duality- a modification
Let $2$ be the discrete topological space with two elements. For a map of sets
$$\beta : X \times Y \rightarrow 2 $$
We get a topology on $X$ and a topology on $Y$. The topology on $X$ is the weakest ...
user30211
4
votes
1
answer
343
views
A continuous function on the disk without non-tangential limits
By Fatou's theorem every bounded holomorphic function on the unit disk has non-tangential limits almost everywhere on the unit circle $\mathbb T$.
Is there an explicit example of a bounded continuous ...
- 687
5
votes
1
answer
213
views
One-dimensional compacta as projective limits
Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1.
Is it possible to express $X$ as a filtering projective limit of finite graphs?
Here finite graphs ...
- 53
5
votes
2
answers
535
views
Collared boundary of a non-metrizable manifold
For this question a manifold-with-boundary is a topological space which is Hausdorff and locally upper-Euclidean. Every metrizable manifold-with-boundary has a collared boundary, as shown in "Locally ...
- 387
16
votes
1
answer
504
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Extending a map from $S^n\to M^n$ to a nice map from $B^{n+1}\to M^n$
Let $S^n$ and $B^{n+1}$ be the unit sphere and unit ball in $\mathbb{R}^{n+1}$, and let $M^n$ be a contractible space of dimension $n$.
If necessary, assume that $M^n$ is a contractible simplicial $n$-...
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13
votes
1
answer
696
views
Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$
From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$.
From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.*
My ...
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6
votes
2
answers
405
views
Do mixing homeomorphisms on continua have positive entropy?
I am trying to understand relations between various measures of topological complexity. I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know ...
- 3,055
0
votes
1
answer
145
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About the finished, $\aleph_0$...-compactness
Definitions :
$(E,d)$ a metric space is finished-compact if any covering of $E$ by open, we can extract a finite subcover
$(E,d)$ is $\aleph_0$-compact if for any infinite covering of $E$ by open, we ...
- 3,801
5
votes
1
answer
647
views
Restrictions of a local diffeomorphism
I am wondering if a local diifeomorphism has the following property (prove or disprove):
Let $M,N$ be differentiable manifolds, and $f:M \to N$ be a local diffeomorphism. Suppose $Z$ is a closed ...
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36
votes
1
answer
3k
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Is there a general theory of "compactification"?
In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
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2
votes
1
answer
471
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Topology: what defines (non-trivial) paths as being the same trace (curve)?
I posted this originally at MathSE but haven't had any feedback and hope for better luck here.
I think I know the answer but can't prove it.
Assume that $Y$ is a Hausdorff space and firstly that $p,...
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2
votes
1
answer
152
views
How to define "interior" for the unit arc? [closed]
Let the unit arc be,
$$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$
There is something I found curious about the unit arc which is that,
It has an empty interior viewed as a ...
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6
votes
0
answers
191
views
Spaces where the Banach-Mazur game is undetermined
Let $X$ be a non-empty topological space. The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I
and II play an inning per positive integer. In the $n$-th inning Player I ...
- 551
0
votes
1
answer
212
views
Dense $G_{\delta}$ set with $\sigma$-porous complement is cofinite?
Let $X$ be a separable Banach space and $D\subseteq X$ be a
proper, connected, and dense $G_{\delta}$ subset of $X$,
$X-D$ is $\sigma$-porous.
Then is $X-D$ contained in a finite-dimensional ...
- 4,969
-2
votes
1
answer
128
views
$G$- space is locally compact [closed]
Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
- 451
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votes
0
answers
61
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Specific property of borelian sigma-algebras
Let X be a set and S a sigma-algebra on X.
Let us name borelian sigma-algebra on X a sigma-algebra that is generated by a topology T on X. Given that it is possible for a set X that some sigma-...
- 2,617
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vote
1
answer
191
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$\mathbb{R}$-like spaces [closed]
Let us call a topological space $(X,\tau)$ $\mathbb{R}$-like if it is homogeneous, connected, $T_2$, has a basis consisting of open sets homeomorphic to $X$, and $|X|>1$.
What is an example of an $...
- 45.5k
1
vote
0
answers
30
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The cellularity of the composition of cellular maps (with arbitrary CW decompositioning)
Is the composition of cellular maps cellular?
Related to this, I have another question. (I apologize to asking very similar question.)
Let ${\sf CWcpx}$ be the category of CW complexes and let ${\sf ...
- 33
3
votes
3
answers
301
views
Continuous map on the simplex (applicable to fair division)
Let $g$ be a continuous function from the unit simplex $D(n)$ (with $n$ vertices) into itself, that leaves invariant its vertices, and such that $g$ is not onto: to fix ideas say that $g(D(n))$ does ...
- 31
2
votes
1
answer
110
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Is the composition of cellular maps cellular?
Let $X$, $Y$, $Z$ be topological spaces homeomorphic to CW complexes. And let $f:X\to Y$, $g:Y\to Z$ be cellular maps.
My question is "Is the composition $g \circ f$ cellular map?".
If $Y$ admits ...
- 33
4
votes
0
answers
73
views
universal 0-dimensional separable metric subspaces
Let $\ \mathscr U:=(U\ \delta)\ $ be a separable metric space which is universal for all finite metric spaces, i.e. for every finite metric space $ \mathscr X:=(X\ d)\ $ there
exists an isometric ...
- 4,686