All Questions
1,339 questions with no upvoted or accepted answers
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Injectively rigid spaces
Given a set $X$, is there a topology $\tau$ such that the identity $\text{id}_X$ on $X$ is the only continuous injective self-map?
(This is Joel David Hamkins's recent question in the category $\...
- 48.9k
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178
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Density of subspace with nonlocal/Wentzell boundary condition
Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by $E=\...
- 111
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62
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Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
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67
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Extending an homotopy, knowing the two base functions extend
Let $A\subset B$ be paracompact spaces, and let $C$ be a paracompact space.
Let $f_0,f_1:A\rightarrow C$ be continuous functions. $F:A\times[0,1]\rightarrow C$ a homotopy from $f_0$ to $f_1$. Suppose ...
- 363
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How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?
Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...
- 841
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Defining connectivity between K points on a periodic domain in terms of proximity
THE SITUATION:
Begin by taking a periodic strip of length 2*Pi. Then use a uniform distribution to place K points (x1,…, xk) on the strip by assigning each of them a randomly sampled number. Then ...
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Decomposition which is locally connected
It is possible construct a connected compact metric space $X$ and a continuous decomposition $\mathcal{G}$ of $X$ that satisfies:
1)$X/\mathcal{G}$ is locally connected.
2)If $M$ is a compact ...
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284
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A question about the Leray-Serre spectral sequence
Suppose $F \to E \stackrel{p}{\to} B$ is a fibration with $B$ simply connected. The $E_2^{p,q}$ page of the Leray-Serre spectral sequence is given by $H^p(B;H^q(F))$. Suppose futhermore that $k$ is a ...
- 2,830
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Minimum rank of certain matrices
Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...
- 13.9k
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126
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Partition refinement of a clopen covering in $\Box (\omega+1)^\omega$
Consider $\omega+1$ with the interval topology, that is $U\subseteq (\omega+1)$ is open if and only if $U\subseteq\omega$ or $(\omega+1)\setminus U$ is finite.
We write $(\omega+1)^\omega$ for the ...
- 48.9k
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99
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Name for condition on map of cancellative monoids
Let $M,N$ be cancellative monoids with identity $\epsilon$ and suppose that $k\colon M\rightarrow N$ is a function such that
$k(\epsilon)=\epsilon$
for all $a,b\in M$, there exists $v\in N$ such that ...
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Homotopy injection between the unit ball in the Euclidean n space and an n-dimensional metric AR
Let $D^n$ be the closed unit ball in $\mathbb{R}^n$. Given a compact, $n$-dimensional, AR(Absolute Retract) metric space $X$, must it happen that either $X$ embeds in $D^n$ or $D^n$ embeds in $X$?
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130
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Is $\mathcal{P}(\omega)/(fin)$ with the interval topology path-connected?
Given a poset $(P,\leq)$ the interval topology on $P$ is generated by
$$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$
where $\downarrow x = \{y\in P: y\leq x\}$ and $\...
- 48.9k
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Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION 1 ...
- 7,500
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260
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Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
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128
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Properties of "incomplete finite simplicial complexes"
Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
...
- 7,609
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163
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The category of discontinuous Banach spaces
A banach space is discontinuous if it is isometric to $DC(X)$ for some Hausdorff topological space $X$. ($DC(X)$ is defined here. We denote by $DBan$, the category of all discontinuous ...
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179
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Topological characterisation of loop spaces
Let $\Omega\colon \mathrm{Top}_*\to\mathrm{Top}_*$ be the loop space functor assigning to each pointed topological space $X$ the pointed space consisting of all based continuous maps $S^1\to X$ ...
- 523
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525
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Separability of the space $C(C[0, 1], \mathbb{R})$
Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$.
I am wondering that ...
- 199
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103
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Lower periodic subsets of groups and semigroups
Suppose that $A$ and $B$ are subsets of a group or semigroup. We call $A$ left
upper [resp. lower] $B$-periodic if $BA\subseteq A$
[resp. $A\subseteq BA$]. If $A$ is both left upper and
lower $B$-...
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137
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(The Homotopy type of the) lifting of homeomorphism of Grassmanian
For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space ...
- 356
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173
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Does real analytic imply locally contractible?
The statement is true for complex analytic spaces. I am not sure who proved this result.
I ask the same question in the real case.
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143
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on reductive monoids which are gorenstein
Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore http://www.cmat.edu.uy/cmat/docentes/alvaro/...
- 3,472
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130
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Regarding graphs of continuous functions between zero dimensional spaces
Background for the question: Let for any topological space $B$, $I(B)$ denote the topological space which has the same set of points as of $B$, and the topology is generated by closed and open sets of ...
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91
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Tubular neighbourhood which is nowhere piecewise linear
I recently asked this question.
I think, if the following were true, then I would solve my problem.
Let $E\subset\{(x_1,\dots,x_n)\in\mathbb R^n\;|\;x_i\geq 0\, \&\, \sum_ix_i=1\}$ be a convex ...
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331
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Relationship between weak Lp and strong Lq topologies for q<p
Specificaly:
Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?
Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not,...
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83
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Topologies on spaces of linear sections
Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable.
Let $f : X \...
- 8,512
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75
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Collapsing a countable collection of intervals on $\mathbb{S}^1$
Consider a countable collection $I_n$ of closed connected disjoint intervals on $\mathbb{S}^1$. When this collection is maximal, the set $\bigcap \nolimits_{i=1}^{n}( \mathbb{S}^1 \backslash \bigcup \...
- 541
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167
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How many ways we have to prove that a topologically (or analytically) nice mapping is injective?
I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and $\...
- 1,881
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479
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Comparing two metrics on the space of infinite sequences and relating open and closed sets
Let $X = \{ 0, 1 \}$ and $X^{\mathbb N_0} = \{ x_0 x_1 x_2 \ldots : x_i \in X \}$ be the space of all infinite sequences, then a metric could be defined on it
$$
d(u,v) := \frac{1}{2^r} \mbox{ with } ...
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178
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Proving that two given functionally structured spaces are isomorphic
The relevant definitions are listed below. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups; and Section 2, Chapter II of Bredon's Topology and ...
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70
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Orbit spaces of involutions on spheres
I'm studying the following problem: Let
$({\mathbb S}^N,\theta)$ be the $n$-sphere (in ${\mathbb R}^{N+1}$) endowed with the antipodal action $\theta:(x_0,\ldots,x_N)\to (-x_0,\ldots,-x_N)$;
$({\...
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130
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A question on star $\sigma$-compact spaces
The question is also posted here.
A topological space $X$ is said to be star $\sigma$-compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a $\sigma$-compact subspace $K$ of $X$ such ...
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275
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Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
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145
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Reference needed: Does pseudo laminated compact subsets of the plane separate the plane?
Hi,
doing my research I found the following problem and I´ll be glad if someone could give a reference.
We say that a compact connected subset $K$ of the plane is psuedo laminated if the following ...
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321
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Type I subspaces of the Stone Cech compactification of $\omega$
EDIT: I found a construction, see below. I decided not to delete the question in case someone is interested.
A space $X$ is of Type I if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ ...
- 1,855
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430
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Intersection of cocompact closed normal subgroups
Let $G$ be a locally compact Hausdorff topological group.
Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology.
Note that ...
- 1,498
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264
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Z-sets in the Hilbert cube
If $(X,d)$ is a metric space, then we say that a closed subset $A$ of $X$ is a z-set if for each number $k\gt 0$ there is a continuous map $f_k$ from $X$ into $X-A$ such that $d(x,f_k(x))\lt k$.
I ...
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245
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Sums of Strongly z-ideals
In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} \mathcal{M_{\alpha}}$...
- 1,788
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Algebraic properties of the semiring of open subsets.
Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?...
- 3,513
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315
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Non trivial definition of bicontinuous functions and the ring of all bicontinuous functions.
At first let me recall that if There are two topology $\tau_1$and $\tau_2$ on a set $X$, the triple $(X,\tau_1,\tau_2)$ is called a bitopological space.
There are many definitions and properties ...
- 1,788
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365
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Killing homotopy groups by removing subsets
Let $X$ be a locally finite CW-complex and let $U$ be an open subset of $X$. Given a non-zero homotopy class $x\in\pi_i(U)$ say, is it possible to find a closed subset $Z\subset U$ whose removal from $...
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Soft sheaves on indiscrete paracompact spaces
Let $X$ be some space, I have basically 2 questions:
1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
- 417
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150
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Follow up question on the measure of the difference between a partial selector and a selector...
This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble...
In Kharazishvili's "Nonmeasurable Sets and ...
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220
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Extension of homeomorphisms on a spherical space
Call a "blot" set, which is the closure of its interior, the boundary is locally connected, and when you remove boundary blot remains connected. Suppose that there is a blot on the surface of the n-...
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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 ...
- 48.9k
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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-...
- 1,211
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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 ...
- 1,402
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61
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Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
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128
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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 ...
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