Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,379
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2
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Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?
As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
- 1,111
2
votes
1
answer
137
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Is the collapse of a totally disconnected compact Hausdorff space still totally disconnected?
Let $S$ be a totally disconnected compact Hausdorff space and let $A\subset S$ be a closed subset. Let $S/A$ denote the space we get when collapsing $A$ to a point. Is this space still totally ...
- 1,900
3
votes
0
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219
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"weakly functorial resolution" of quasi-compact T_1 topological space by quasi-compact Hausdorff space
I have an arguably weird question: Let $X$ be a quasi-compact $T_1$ topological space, could there be a construction that takes such an $X$ as input and outputs a surjection
$$X' \to X$$
with the ...
- 619
5
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0
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234
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Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 619
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1
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74
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Extending maps from a discrete set to a Stone-Čech compactification while retaining an injectivity condition
For $S$ a set, let $\beta_{\bf2}(S)$ be a compact, totally disconnected space containing $S$ where $S$ in the subspace topology is discrete and $S$ is a dense subspace, and $\beta_{\bf2}(S)$ has the ...
- 1,250
3
votes
2
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264
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Cut a homotopy in two via a "frontier"
Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$.
(...
5
votes
0
answers
122
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Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?
Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
- 59k
1
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1
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289
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Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 167
10
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0
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229
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Arhangel'skii's problem revisited
One of the most well-known problems in set-theoretic topology is Arhangel'skii's question of whether there exists a Lindelöf Hausdorff space with "points $G_\delta$" (meaning, every point is ...
- 4,283
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75
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Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 1,958
4
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1
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469
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Do CGWH spaces form an exponential ideal in Condensed Sets?
If $X$ is any condensed set and $Y$ is a compactly generated weak Hausdorff (CGWH) space (a.k.a. $k$-Hausdorff $k$-space), is $Y^X$ again a CGWH space? To be more precise, is $(\:\underline{Y}\,)^X$ ...
- 141
1
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0
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104
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Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
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0
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73
views
Can we construct general counterexample to support the Weak Whitney theorem? [duplicate]
Can we construct an example for the weak Whitney theorem to illustrate the existence of a continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold that cannot be smoothly ...
- 109
3
votes
0
answers
189
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Contractibility of the pseudo-boundary of the Hilbert cube
Let the separable Hilbert cube $Q=\prod_{i=1}^{+\infty}[0,1]$ embed into the real Hilbert space $H=l^2(\mathbb{Z}^+)$, whose coordinate unit vectors are $\{ e_i \}_{i=1}^{+\infty}$, as the subset $\...
- 1,069
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147
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Gluing faces of n-cube
Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$.
Let $f_0$ and $f_1$ be faces ...
- 63
33
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6
answers
4k
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Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 717
4
votes
1
answer
146
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A "simple" space with closed retracts but non-unique sequential limits
This question asking how KC ("Kompacts are Closed") and RC ("Retracts are Closed") are distinct has some good discussion, including a now-published example by Banakh and Stelmakh ...
- 933
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0
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58
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A cellular automaton with an image that is not closed
Let $G$ be a non-locally finite periodic group and let $V$ be an infinite-dimensional vector space over a field $\mathbb{F}$. Does there exist a nontrivial topology on $V^G$ and a linear cellular ...
- 885
2
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1
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212
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Is the projective limit $\mathcal{D}(\mathbb{R})$ separable?
Let $\mathscr{D}(\mathbb{R})$ be the set $C_0^\infty(\mathbb{R})$ of smooth functions with compact support endowed with the following topology:
The initial topology with respect to the family maps $(\...
3
votes
1
answer
279
views
Pointwise convergence and disjoint sequences in $C(K)$
Let $K$ be a Hausdorff compact space and let $C(K)$ be the space of continuous real-valued functions on $K$. A sequence $(h_n)$ in $C(K)$ is called almost disjoint if there is a sequence $(g_n)$ with ...
- 5,275
6
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0
answers
177
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Conditions for metrisability
If a normal, first countable space is the union of countably many open metrisable subspaces, must that space be metrisable?
Partial answers, which I proved in the 1980's, include:
(0) The answer is ...
- 69
7
votes
1
answer
298
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G-topological spaces and locales
Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...
- 91
0
votes
0
answers
15
views
Is the impression of an ideal boundary point (=end) the union of the impressions of the prime ends of the circle of prime ends associated to this end?
Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (...
0
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0
answers
37
views
Generic non-existence of 1. Integral of continuous DS
Let $M$ be a compact manfiold and $F \in \mathcal{X}(M)$. We define a DS on $M$ by
$$\dot{\mathbf{x}}=F(\mathbf{x}(t))$$
In 1 it was shown by Hurley, that a generic diffeomorphism on $M$ does not have ...
- 227
8
votes
1
answer
318
views
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
- 725
3
votes
1
answer
144
views
Embedding of half open half closed $n$-set in $n$-space
Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...
2
votes
1
answer
126
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Variation of concept of a Lusin space
Citing from Wikipedia,
A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space.
Is there a (previously studied) analogous concept of a Hausdorff (...
- 611
1
vote
1
answer
598
views
Cohomology of the amplitude space of unlabeled quantum networks
I am investigating a particular map from a product of three-spheres to the moduli space of (non-negative, real edge weight) networks. The map in question is
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...
- 519
1
vote
1
answer
75
views
Reference for k-Hausdorff (in terms of compact T2 images)
In Rezk - Compactly generated spaces a k-Hausdorff property is defined, between weakly Hausdorff and unique sequential limits.
On the other hand, a stronger notion of k-Hausdorff between $T_2$ and ...
- 933
14
votes
0
answers
405
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Which functions have all the common $\forall\exists$-properties of continuous functions?
This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well.
For a ...
- 19k
6
votes
1
answer
430
views
Which maps of topological spaces have the right lifting property with respect to all split monomorphisms?
Let $p : X \to Y$ be a continuous map. We say that $p$ has the right lifting property with respect to split monomorphisms if, for every space $B$, and every retract $A \subseteq B$, and for every ...
- 59k
0
votes
0
answers
290
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Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
- 227
0
votes
1
answer
443
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
- 167
8
votes
1
answer
138
views
Stone-topological/profinite equivalence for quandles
A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:
(Q1) ...
- 253
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0
answers
41
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
- 595
10
votes
1
answer
295
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A topological characterisation of a.e. continuity
We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
- 4,232
2
votes
1
answer
115
views
Homeomorphisms of the projective cover of the Cantor set
Let $M$ be the projective cover (e.g, Gleason1958) of the Cantor set $\{-1,1\}^{\mathbb{N}}$. Let $\textrm{homeo}(M)$ denote the group of all homeomorphisms of $M$.
Some of the $\gamma\in\textrm{homeo}...
- 2,118
2
votes
1
answer
119
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Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product
I've been trying to understand various questions to do with sigma algebras on uncountable product spaces.
Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
- 1,111
0
votes
0
answers
94
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 280
9
votes
1
answer
420
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Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?
We work in ZFC.
Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$.
A field $E$ is ...
- 635
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votes
2
answers
254
views
Must US extremally disconnected spaces be sequentially discrete?
Based upon discussion at Math.SE
Consider the property extremally disconnected, for which the closure of any open set remains open.
Frequently, this property is paired with the assumption of Hausdorff....
- 933
1
vote
1
answer
110
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Extremally disconnected rigid infinite Hausdorff compacta(?)
Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism
$\ h: X\to X\ $ is the identity homeomorphism
$\ h=\mathbb I_X:\ X\to X\...
- 4,674
1
vote
1
answer
72
views
Subspaces generated by the orbits of the group of isometries on $C(K)$
Let $X$ be an extremally disconnected compact Hausdorff space with no open points, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D_f$ be the linear span of the functions of the ...
- 2,118
11
votes
4
answers
2k
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Early illustrations of topological notions in published work
Cross-posted from HSM: I posted this question a bit more than a week ago but have not gotten any answers at HSM. The only comment on the posting asks if I would accept polyhedral pictures ...
- 24.5k
0
votes
0
answers
69
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Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
- 227
1
vote
2
answers
190
views
A few questions about Tychonoff plank
In the Morita's following article (K. Morita. Some properties of M-spaces), constructing an space $X$ and defining an identification on it.
My first question is how to prove that $S$ is countably ...
- 1,109
0
votes
1
answer
138
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Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of ...
- 1,111
3
votes
1
answer
246
views
Can such a set be simply connected?
$\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) ...
- 111k
14
votes
1
answer
411
views
Topology and pcf theory
$\DeclareMathOperator\pcf{pcf}$For simplicity say $\aleph_\omega$ is a strong limit. Let $A=\pcf\{\aleph_n:n\in\omega\}$. Then it follows from basic properties of pcf operation that $X\subseteq A\...
- 405
5
votes
2
answers
500
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On the boundary of a simply connected set
Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x ...
- 111k