All Questions
Tagged with general-topology or gn.general-topology
4,600 questions
3
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0
answers
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Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?
This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested:
Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
- 2,830
3
votes
1
answer
352
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Sequential separability on $C_p(X)$
Definition. Let $E$ be a topological space. Suppose that $E$ contains a sequence $\{x_n\}$ such that for every $x\in E$, there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x=\lim x_{n_k}$. ...
- 4,713
7
votes
0
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184
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Maps with small fibers between manifolds of equal dimension
The following question is an attempt to revise this one into what I intended.
Important revisions are shown in bold.
Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
- 781
5
votes
1
answer
279
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Codimension zero embeddings and maps with small fibers
Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.
...
- 12.9k
13
votes
1
answer
839
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Mistake on article about Bohr compactification?
$\DeclareMathOperator\b{b}\newcommand\B{{\operatorname B}}$I wish to get help understanding the content of two theorems of [Iva] that seem mutually contradictory. First some context. Let $\b(\mathbb{R}...
- 11.4k
12
votes
1
answer
385
views
Is $X\times X$ homeomorphic to $X$ for a space of probability measures?
Let $\mathcal M_1(S)$ be the (compact, metrizable) space of probability Borel measures on the circle $S=\{z\in\mathbb C: |z|=1\}$ with its weak $*$ topology, so $\mu_n\to\mu$ if and only if
$$
\int_S ...
- 4,461
7
votes
1
answer
291
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Lower bound on dimension required to disconnect manifold?
This question seems quite classical, but I don't quite know what subarea of topology it falls into.
Suppose that removing the set $S$ disconnects the 2-torus $\mathbb{T}^2 = \mathbb{R}^2\diagup\mathbb{...
- 28.7k
3
votes
1
answer
144
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Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$.
Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that
$$
\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0
$$ as $r\to0$?
If you like,...
- 6,357
2
votes
0
answers
157
views
About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
- 627
15
votes
3
answers
3k
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Category of topological spaces with open or closed maps
Consider the category whose objects are topological spaces and whose morphisms are the open maps (or closed maps, open continuous maps, closed continuous maps … that is, one whose isomorphisms are ...
- 12.9k
3
votes
1
answer
342
views
Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$
The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
- 7,193
5
votes
1
answer
92
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Preimage of a sublocale by a morphism of locales: description by nucleus?
For completeness of MathOverflow, and to avoid any possible misunderstanding, let me recall the following terminology and facts, which should be standard (experts skip the following 2–3 paragraphs ...
- 42.4k
9
votes
1
answer
294
views
Connected open sets in the topology generated by the collection of connected open sets
Let $(X,\mathcal{T})$ be a connected topological space. Let $\mathcal{T}'$ be the topology on $X$ that is generated by the collection of connected open sets in $(X,\mathcal{T})$. That is, the ...
- 716
10
votes
1
answer
657
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 66.3k
3
votes
1
answer
90
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Even covers and collectionwise normal spaces
We call $X$ strongly collectionwise normal if the set $\mathcal{U}_\Delta$ of all neighbourhoods of the diagonal $\Delta_X$ of $X\times X$ is a uniformity. This is equivalent to the property that for ...
- 716
1
vote
1
answer
379
views
Creating an inverse system which "stratifies density"
Setting:
Let $X'$ be a dense subset of an infinite-dimensional Fréchet space $X$ and suppose that $(X_n')_{n \in \mathbb{N}}$ is a nested sequence of non-empty subsets of $X'$ satisfying
$$
\bigcup_{n ...
CommunityBot
- 1
3
votes
1
answer
130
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 2,830
14
votes
1
answer
496
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Is there an 'unnatural' topological construction of an algebraically closed field of positive characteristic?
It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ...
- 118k
1
vote
0
answers
41
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Why does the Kieboom characterization of shape is restricted only to paracompact spaces?
Borsuk founded shape theory as an extension of homotopy theory, appropriate for spaces with bad local properties. Borsuks definition was applied only to compact metric spaces. Later, this was ...
- 111
0
votes
0
answers
62
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Order-convergence and interval topology on ${\cal P}(\omega)/(\text{fin})$
On any poset $(P, \leq)$ we can consider two different topologies that arise directly from the ordering relation.
1) Order convergence topolog $\tau_o(P)$ : By a set filter $\mathcal{F}$ on $P$ we ...
- 48.8k
0
votes
1
answer
99
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A question about G-Hewitt spaces
In the paper linked below, S. A. Antonyan gives the following proposition without proof (in fact all results are given without proof). I need a proof of this theorem. If anyone has information on this ...
- 11.4k
3
votes
2
answers
340
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Cohomology version of Moore space
I asked this question on MSE a few days back but could not get any helpful response. So I am rewriting the post.
It is known to me that given a simply connected finite dimensional (which is also level-...
- 10.8k
8
votes
1
answer
1k
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What's the point of a point-free locale?
In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...
- 32.5k
1
vote
2
answers
202
views
Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications:
Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
- 1,354
3
votes
3
answers
255
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Continuum-distanced complete, ultrametric space
Our professor asked us to find a complete metric space where the intersection of nested closed balls can be empty.
The following space is such an example, and I would like to learn more on it (since ...
- 4,786
0
votes
0
answers
43
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Equivalent conditions for $z$-embeddability
I am looking for where this specific theorem of Blair is originally located:
Theorem. Let $S\subseteq X$, the following are equivalent:
$S$ is $z$-embedded
If $A, B\subseteq S$ are disjoint zero-...
- 1,201
3
votes
1
answer
247
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Relation between $\mathbb{R}$ and the metric space of bounded functions $f:\mathbb{N}\to\mathbb{N}$
Let $\newcommand{\N}{\mathbb{N}}\newcommand{\B}{\mathbf{B}}\B(\N)$ be the collection of all bounded functions $f:\N\to\N$. (A function $f:\N\to\N$ is bounded if there is $M\in\N$ such that $f(k) < ...
- 11.4k
3
votes
1
answer
395
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Closed embedding into a normal Hausdorff space and left lifting property
I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a ...
CommunityBot
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1
vote
0
answers
68
views
"Bad" valid edge contractions
In this paper, an edge contraction of a simplicial complex $\Gamma$ is defined as the operation of removing the neighborhood $N_e\Gamma$ of the edge $e=\{0,1\}$ and identifying $N_0\partial N_e\Gamma$ ...
- 11
2
votes
1
answer
131
views
Strong ultralimits?
I was going through the book Ultrafilters Throughout Mathematics and I came across the notion of ultralimits, defined below.
Ultralimit. Let $(X,\tau)$ be a topological space, $(x_i)_{i\in I}$ be a ...
- 11.4k
1
vote
2
answers
127
views
Homeomorphism and boundary of a complementary component
Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.
More precisely, let $h:X\to X$ be a homeomorphism.
...
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votes
2
answers
314
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Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
- 1,201
5
votes
1
answer
2k
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Proof that the Pontryagin dual of a topological group is a topological group
I'm looking for a proof that the Pontryagin dual $G^*$ of a topological group $G$ is a topological group.
It's very easy to prove that $G^*$ is a group, my troubles are in proving that the map $G^* \...
- 105k
3
votes
1
answer
136
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For $\mathbb R^n \times Q \cong \mathbb R^m \times Q $ must $n = m$? ($Q$ is the Hilbert cube)
There are several theorems describing the topology on hyperspaces of convex subsets of $\mathbb R^n$ under the Hausdorff metric. For example Antonyan and Jonard-Pérez prove the space of compact convex ...
- 11.1k
0
votes
0
answers
42
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Topologizing quasi orders with regards to products
This morning I was asked by a colleague for the "right" way to construct a topology on a quasi-order (aka preorder, a reflexive and transitive relation) such that the topology on a product ...
- 1,322
0
votes
1
answer
98
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Is every subgroup closed in this complete, nondiscrete topological group?
Another question on Mathoverflow (here: Complete topological groups in which all subgroups are closed) asks if there exists a complete, nondiscrete topological group $G$ such that all subgroups of $G$...
- 11.4k
3
votes
1
answer
1k
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"Relative compactness of a family of probability measures" and relative compactness & sequential compactness of sets
I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable.
In the discussion ...
CommunityBot
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7
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1
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384
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Compact Hausdorff spaces as a cocompletion of profinite sets
It is well-known that the category CH of compact Hausdorff spaces has a strong categorical flavor (e.g. Properties of the category of compact Hausdorff spaces, which includes Manes' theorem asserting ...
- 12.9k
8
votes
1
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468
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Finite domination and compact ENRs
Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only ...
- 5,596
7
votes
2
answers
529
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What is the name for a point that is periodic to within $\varepsilon$?
Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...
- 5,407
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1
answer
198
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Topological property of the space of probability measures
Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence.
Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the ...
- 24.2k
4
votes
0
answers
47
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Are W-spaces with countable pseudocharacter first countable?
Cross-post of a question originally asked by Almanzoris on Mathematics Stack Exchange.
A topological space $X$ is called W-space if P1 has a winning strategy at each point $x \in X$ for the following ...
- 1,322
2
votes
1
answer
103
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LCH spaces $X$ such that if $Y$ is a perfect image of $X$, then $Y$ is zero-dimensional
I am looking for locally compact Hausdorff spaces $X$ with the following property:
If $f:X\to Y$ is a perfect map onto locally compact Hausdorff space $Y$, then $Y$ is zero-dimensional.
One can see ...
- 11.4k
0
votes
0
answers
32
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Hausdorff dimension: The dimension of boundary of a set [migrated]
I can't understand the following statement.
If (perhaps not closed) set $S$ has dimension $n$, then the boundary could have any dimension from $0$ to $n$. (Could someone give me an example?)
If S ...
- 129
7
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0
answers
270
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Generalizing uniform structures as Grothendieck topologies
Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
- 519
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1
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86
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Lattice of functions and their minimal separating set upto topological equivalence
There is a very wide series of questions I have been thinking about and I am wondering if there is any literature on this type of structures.
Let's start with the set of all functions $F: \mathbb{R} \...
- 39
9
votes
1
answer
424
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Delta-generated spaces vs CW complexes
$\newcommand\Top{\mathrm{Top}}\newcommand\CW{\mathrm{CW}}\newcommand\Deltagenerated{\text{$\Delta$-generated}}\newcommand\Spaces{\mathrm{Spaces}}\newcommand\DeltaSpaces{\text{$\Delta$-Spaces}}$I am ...
- 7,193
2
votes
0
answers
406
views
Complete topological groups in which all subgroups are closed
My previous question has been answered by YCor; so I am asking a new one with a reasonable additional assumption. See the previous question for the background and motivation.
General question: does ...
- 361
21
votes
1
answer
2k
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Characterization of Fréchet-Urysohn spaces using sequential continuity at a point
A map $f \colon X \to Y$ is called sequentially continuous at the point $a$ if for every sequence $(x_n)$ such that $x_n\to a$, we also have $f(x_n)\to f(a)$.
$$x_n\to a \qquad \Rightarrow \qquad f(...
- 21
7
votes
2
answers
383
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Connectivity of fibers under fibration replacement
Assume all the spaces mentioned below are simply connected CW complexes. Let $ f: X \to Y $ be a continuous surjctive map between CW complexes, where $ f $ is not necessarily a fibration. Assume that ...
- 63.9k