All Questions
Tagged with topology or gn.general-topology
4,599 questions
7
votes
0
answers
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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 ...
- 739
13
votes
1
answer
817
views
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}...
- 193
5
votes
1
answer
275
views
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.
...
- 739
12
votes
1
answer
371
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
$$
\int_S ...
- 24.2k
7
votes
1
answer
289
views
Lower bound on dimension required to disconnect manifold?
This question seems quite classical, but I don't quite know what subarea of topology it falls into.
Suppose that removing the set $S$ disconnects the 2-torus $\mathbb{T}^2 = \mathbb{R}^2\diagup\mathbb{...
- 2,621
3
votes
1
answer
143
views
Jordan plane curve such that $\frac{d(g(x),g(y))}{d(x,y)}\to0$?
Write $g$ as the inverse of $f$.
Is there a continuous injective $f:S^1\to C\subset\mathbb{R}^2$ such that
$$
\displaystyle\sup_{d(x,y)<r}\dfrac{d(g(x),g(y))}{d(x,y)}\to0
$$ as $r\to0$?
If you like,...
- 211
2
votes
0
answers
156
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About the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev
According to the 7.3.5. Corollary of the book "Measure Theory" by V.I. Bogachev we have the following result:
Let $(X,\tau)$ be a completely regular space and let $\Gamma$ be a
family of ...
- 627
3
votes
1
answer
340
views
Fundamental group of the grid on $\mathbb{R}^\mathbb{N}$
The grid on $\mathbb{R}^2$ is defined by the set of points such that at most one coordinate is not an integer. With this in mind, e endow $\mathbb{R}^\mathbb{N}$ with the product topology, where $\...
- 48.8k
9
votes
1
answer
293
views
Connected open sets in the topology generated by the collection of connected open sets
Let $(X,\mathcal{T})$ be a connected topological space. Let $\mathcal{T}'$ be the topology on $X$ that is generated by the collection of connected open sets in $(X,\mathcal{T})$. That is, the ...
10
votes
1
answer
654
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 168
3
votes
1
answer
128
views
Is the interval topology on ${\cal P}(\omega)/(\text{fin})$ connected?
If $(P,\leq)$ is a poset and $x\in X$, we let $\downarrow x = \{p\in P: p \leq x\}$, and $\uparrow x$ is defined dually. The collection $$\Big\{P\setminus (\downarrow x): x\in P\Big\} \cup \Big\{P\...
- 48.8k
3
votes
1
answer
90
views
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 ...
- 1,201
1
vote
0
answers
41
views
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
14
votes
1
answer
491
views
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 ...
- 12.4k
0
votes
0
answers
62
views
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
5
votes
1
answer
91
views
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 ...
- 32.4k
0
votes
1
answer
99
views
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 ...
- 1,367
3
votes
2
answers
339
views
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-...
- 1,177
0
votes
0
answers
43
views
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
3
answers
255
views
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 ...
- 637
1
vote
0
answers
67
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 ...
- 23
3
votes
1
answer
247
views
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) < ...
- 48.8k
1
vote
2
answers
125
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.
...
- 3,317
3
votes
1
answer
136
views
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 ...
- 1,955
0
votes
0
answers
42
views
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
8
votes
1
answer
197
views
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 ...
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 $\beta X\setminus X$ ...
- 1,201
4
votes
0
answers
47
views
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
views
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 ...
- 1,201
0
votes
0
answers
32
views
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 ...
- 29
9
votes
1
answer
424
views
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 ...
- 521
11
votes
2
answers
314
views
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
0
votes
1
answer
98
views
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 $G$...
- 361
7
votes
2
answers
383
views
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 ...
- 1,177
1
vote
0
answers
104
views
Commutative Banach $\mathbb{R}$-algebras without complex structure, but with path-connected group of units
For a finite-dimensional commutative (associative, unital) $\mathbb{R}$-algebra $A$, the condition $\pi_0(A^\times) = 1$ (i.e. the group of units of $A$ being path-connected) is equivalent to $A$ ...
- 7,127
0
votes
1
answer
79
views
Dimension of a manifold derived from a dense $G_{\delta}$ subspace
Let $X,Y$ be (compact connected) topological manifolds of dimensions $n,m$, respectively. Assume that a dense $G_{\delta}$ subspace $A$ of $X$ is homeomorphic to a dense $G_{\delta}$ subspace $B$ of $...
4
votes
0
answers
97
views
Is there a concept of a map of Grothendieck sites having dense image?
Someone recently asked if one can talk about a map being etale dense just like one can talk about it being Zariski dense. My main question is: has anyone discussed such a notion?
On a simple ...
- 15.4k
1
vote
1
answer
91
views
When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
- 605
7
votes
1
answer
339
views
Is $C(X, \{0,1\})$ locally compact?
Let $X$ be a locally compact Hausdorff space. Let $C(X, \{0,1\})$ be the space of continuous functions $X \to \{0,1\}$ with the compact-open topology, that is, the topology generated by the following ...
- 1,485
3
votes
0
answers
75
views
Can we generalize the Kuratowski Extension Theorem to Souslin spaces?
The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\...
- 627
2
votes
2
answers
154
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Closure of $C([0,1]^2)$ via weak*-topology [closed]
Let $C([0,1]^2)$ denote the set of continuous functions on $[0,1]^2$. Let $L^1([0,1]^2)$ be the set of all Lebesgue integrable functions on $[0,1]^2$.
The dual space of $C([0,1]^2)$, denoted by $C^*([...
- 349
9
votes
1
answer
625
views
The reals: a topological lattice in more than the obvious way?
Define a topological lattice as a (not necessarily bounded) lattice in $\textbf{Top}$, i.e. meet and join are continuous maps $X^2 \rightarrow X$. There are two obvious topological lattice structures ...
- 591
0
votes
1
answer
230
views
Questions on the compactness of $L_1([0,1]^2)$'s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$ and $f(x,y)\geq 0: a.e. (x,y)\in [0,1]^2$. Recently in my study I need to study the compactness of $U$. By Riesz's theorem ...
- 349
0
votes
1
answer
99
views
Embeddings of pseudo metric spaces into seminormed Spaces
There is a theorem stating that every metric space embeds isometrically into $\ell _{\infty}$.
My question: is there a generalized result for pseudo metric spaces embedding isometrically into semi-...
- 85
0
votes
1
answer
100
views
Limit sequence of regular function in $L_1$‘s unit sphere
Let $U$ denote the set of functions $f\in L_1([0,1]^2)$ such that $\int f=1$. For any $f\in U$, we say it is regular if $\int_{x_0\times [0,1]}f=\int_{[0,1]\times y_0}f=1$ for a.e. every $x_0, y_0\in [...
- 349
0
votes
0
answers
128
views
The smallest dihedral angle of convex polyhedrons
Given a set of points $\{x_{k}\}_{k=0}^{m} \subset \mathbb{R}^n$, is it always possible to find a constant $c=c(m,n)>0$, depending only on the dimension $n$ and the number $m$, such that, after ...
- 695
7
votes
2
answers
296
views
Compactly generated and paracompact $\Rightarrow$ Hausdorff?
In A Concise Course in Algebraic Topology by May, a proposition is stated that any open cover of a paracompact space has a numerable refinement, where the space is assumed to be compactly generated ...
-1
votes
1
answer
167
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 349
2
votes
0
answers
29
views
When are canonical maps of a filtered colimit open/closed, given that the transition maps are open/closed?
Let $X_i$ be a filtered diagram of topological spaces. I am interested in when the canonical maps $f_i:X_i\rightarrow \text{colim } X_i$ are open/closed. It is pretty easy to show that if the ...
- 41