All Questions
5,184 questions
0
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2
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673
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(Homotopy) Y ENR and contractible subset implies Y is a retract
I'm trying to solve the following question:
Suppose $Y \subset R^n$ is a Euclidean neighborhood retract. I want to prove that if $Y$ is contractible, then it is a retract of $R^n$.
- 11
0
votes
1
answer
339
views
Thurston-Bennequin number vs. checkerboard coloring difference
For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1
kinks exist (K has ...
- 4,829
0
votes
1
answer
277
views
Diffeomorphisms of a surface in terms of generators.
I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism (...
- 192
0
votes
2
answers
641
views
Looking for general approaches to show connectedness of topological groups
Let $G$ be a topological group. One general approach to show that $G$ is connected is the following:
For every subgroup $H\leq G$ (not necessarily closed) we have a projection map:
$$
\pi: G\...
- 7,609
0
votes
1
answer
232
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 ...
- 359
0
votes
1
answer
101
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 [...
- 359
0
votes
1
answer
60
views
Hereditary property of bionto (bi-onto) functions
Let $\,\ X\,\ Y\ $ be topological spaces. Set $\ A\subseteq X\ $ is said to be clopen in $\ X\ $ iff both $\ A\ $ and $\ X\setminus A\ $ are open. Continuous function $\ f:X\to Y\ $ is said to be ...
- 4,786
0
votes
1
answer
92
views
Continuous selectors of a continuous multifunctin on a compact metric space
I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
...
- 79
0
votes
1
answer
109
views
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 ...
- 1,644
0
votes
1
answer
254
views
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 ...
- 825
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votes
1
answer
101
views
A question on relation of different triangulations of a triangulable space
Suppose we get two triangulations of a manifold with boundary $M$ such that the triangulation is compatible with boundary, i.e. the restriction on the boundary is itself a triangulation, is it these ...
- 781
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1
answer
199
views
Is the Čech–Stone compactification of the integers always a retract of an extremally disconnected space?
Probably $\beta \mathbb N$ is not an absolute retract (is there an easy argument for this?), but I'd be interested to know what happens in the class of extremally disconnected (compact) spaces. Is it ...
- 11.3k
0
votes
1
answer
279
views
When does strict inclusion holds for the domain of subdifferential?
Recall that, given an extended real-valued function $f: \mathbb{R}^n \to (-\infty, \infty]$
Its effective domain is,
$$\text{dom}(f) = \{x \in \mathbb{R}^n : f(x) < +\infty\}$$
The subdifferential ...
- 103
0
votes
1
answer
103
views
(Seeking Definition) What Does it Mean for a Space to have Rim-Type $\alpha$? Or the 'derivative' of a Countable Set?
I've encountered a definition in several papers, but literally none of them define the term. They all instead reference a book by Menger that has never been printed in English. The term is "rim-...
- 439
0
votes
1
answer
100
views
Connectedness of the set having a fixed distance from a closed set 2
This question is related to this one: Connectedness of the set having a fixed distance from a closed set. Suppose $F$ is a closed and connected set in $\mathbb{R}^n$ ($n>1$). Suppose the complement ...
- 411
0
votes
1
answer
170
views
P-filter property?
Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup_i ...
- 1,101
0
votes
1
answer
324
views
Is a totally ordered, separable and connected topological space metrizable (in the order topology)?
Is a totally ordered, separable and connected topological space metrizable (in the order topology)?
If we relax the assumption of connectedness, I know the counterexamples, but if we have a linear ...
- 327
0
votes
1
answer
148
views
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 ...
- 4,074
0
votes
1
answer
203
views
Filtered colimit of a topological space
Suppose that $X$ is a space filtered by closed subspaces $X_{1}\subset X_{2}\subset \dots$.
As topological space $X=\operatorname{colim}_{n}X_{n}$.
We define $Y_{n}=X_{n+1}/X_{n}$, and consider the ...
- 511
0
votes
1
answer
89
views
Morphism of schemes with non-sober image
Let $f:X\rightarrow Y$ be a morphism of schemes. Can the image of $f$ endowed with the subspace topology not be sober?
user138661
0
votes
1
answer
152
views
Reference request: Baire class 2 functions
There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
- 2,069
0
votes
1
answer
208
views
strict topology on multiplier algebras
Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...
- 451
0
votes
1
answer
273
views
Continuity of maps in which preimage preserves compactness
Let $X$ and $Y$ be Hausdorff spaces and suppose that $Y$ is locally compact. Let $f:X\to Y$ be a surjective map such that for any compact subset $K \subset Y$ the pre-image $$f^{-1}(K)=\{x\in X: f(x)\...
- 563
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1
answer
282
views
Does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and …?
Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with non-zero finite length $L$. Then, does there always exist a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
- 113
0
votes
3
answers
554
views
Converting a bounded metric into an unbounded metric
Suppose $d$ is a bounded metric on $X$, i.e. $d(x,y)< K<\infty$ for all $x,y\in X$. Is there a standard way to convert $d$ into another metric $\widetilde{d}$ on $X$ with the property that $\...
- 710
0
votes
1
answer
129
views
Is there a $\sigma$-metacompact space which is not metacompact?
Recall that a space $X$ is metaLindelof if every open cover of
$X$ has a point-countable open refinement.
A space $X$ is metacompact if every open cover of
$X$ has a point-finite open refinement....
- 621
0
votes
1
answer
103
views
Continuous orthogonal preserving maps between projective space
Is there a continuous map $f:\mathbb{C}P^n \to \mathbb{C}P^m$, for some $n>m$
which preserve orthogonality?Namely $x\perp y \implies f(x) \perp f(y) $?
If yes, are there two non homotopic ...
- 356
0
votes
1
answer
668
views
A possible proof of the Borsuk Ulam theorem without "Homology-Cohomology"
Assume that $n>1$.
The configuration space of $S^n$ is defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$
We have two questions:
1.Is there a continuous function $f:M_n ...
- 356
0
votes
1
answer
57
views
Refining ultraconnected spaces to connected $T_2$ spaces
Let $X$ be an infinite set and suppose $\tau$ is an ultraconnected topology on $X$ without isolated points. Is there a topology $\sigma\supseteq \tau$ such that $(X,\sigma)$ is a connected $T_2$-space?...
- 48.9k
0
votes
1
answer
74
views
$T_2$-spaces such that the lattices of open sets can be embedded into each other
Let $(X,\tau), (Y,\sigma)$ be $T_2$-spaces such that there are injective lattice homomorphisms $f: \tau\to \sigma$ and $g:\sigma\to \tau$.
Does this imply that $(X,\tau)\cong (Y,\sigma)$?
user62017
0
votes
1
answer
384
views
Heisenberg group acts on the circle
Let $H$ be a Heisenberg group, i.e.
$$
H=\left\langle a,b,c |[a,b]=c,[a,c]=[b,c]=1\right\rangle.
$$
$H$ acts on the circle by homeomorphism which preserves the orientation. If the rotation number of $...
- 21
0
votes
1
answer
134
views
Two different products of filters
By filters I will mean filters on some set $\mho$.
I define product of an infinite family of filters in two ways. I feel (by analogy with properties of Tychonoff product vs box product of topological ...
- 765
0
votes
1
answer
237
views
How to see such space is Lindelof?
Let $R$ denote the set of all real numbers. $B$ is any Bernstein set of $R$.
Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains ...
- 654
0
votes
1
answer
232
views
Kernel elements for the Grothendieck group map of a commutative monoid
This is just a nomenclature question. Let $T$ be a commutative monoid, and let $T^*$ be its Grothendieck group. That is, $T^* \cong T \times T \ / \sim$, where $(s,s') \sim (t, t')$ if $s+t'+e = s'+t+...
- 8,512
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votes
3
answers
1k
views
Zero-dimensional space
Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that $\overline{A}=\overline{\...
- 192
0
votes
1
answer
278
views
On the compactness of a certain chain topology [closed]
Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set ...
- 243
0
votes
1
answer
137
views
Connectedness of a union regading a proximity
Let δ is a proximity.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Question: Let A and B are sets with non-empty intersection. Let both A and B ...
- 765
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
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
0
votes
1
answer
89
views
Monto functions (multiply onto functions)
This is an improvement over Hereditary property of bionto (bi-onto) functions.
Let $\,\ X\,\ Y\ $ be topological spaces. Set $\ A\subseteq X\ $ is said to be clopen in $\ X\ $ iff both $\ A\ $ and $\ ...
- 4,786
0
votes
1
answer
135
views
Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
0
votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
80
views
Continuous modification of tangent vector fields
Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
- 434
0
votes
2
answers
287
views
Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
- 612
0
votes
2
answers
182
views
Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)
Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
\mathcal{P}:=\left\{ X = (X_t)_{t \in \mathbb{Z}} \, ...
- 135
0
votes
1
answer
225
views
Does contractible imply homologically locally connected?
Spanier mentions that locally contractible implies homologically locally connected but I'm wondering whether contractible implies homologically locally connected?
Definition of homologically locally ...
0
votes
1
answer
137
views
Examples of b-connected sets?
B is a b-open set if $B\subset Cl(IntB) \cup Int(ClB)$
A topological space $X$ is b-disconnected if it can be expressed as a union of two disjoint non-empty b-open sets. Otherwise, $X$ is said to be ...
0
votes
2
answers
263
views
Finite sheeted covering of the complement of a finite set in $\mathbb{C}$
For figure "eight" there is a list of finite sheeted covering discussed in Hatcher's book "Algebraic topology". I was thinking about the following question:
Let $S$ be a finite ...
- 1,177
0
votes
1
answer
1k
views
What is definition of branched covering?
What is definition of branched covering in the page 10 of following paper ?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...
- 119
0
votes
1
answer
49
views
More vocabulary for periodic elements in monoids
Let $M$ be a monoid, and let $x\in M$. One says that $x$ is periodic if
$$x^{i+j}=x^j$$
for some integers $i\geq 1$ and $j\geq 0$.
An easy division algorithm argument shows that if $m$ is the ...
- 18.7k