**3**

votes

**0**answers

101 views

### Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a ...

**1**

vote

**0**answers

47 views

### 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 ...

**11**

votes

**1**answer

230 views

### On the global structure of the Gromov-Hausdorff metric space

This is a purely idle question, which emerged during a conversation with a friend about what is (not) known about the space of compact metric spaces. I originally asked this question at ...

**2**

votes

**0**answers

69 views

### cross-sections of a sphere bundle

Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...

**1**

vote

**0**answers

35 views

### 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 $X_1$.
...

**4**

votes

**0**answers

58 views

### Surjectivity of self-isometries as property of metric spaces

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...

**0**

votes

**1**answer

64 views

### Bounded metric spaces with non-surjective self-isometry

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...

**4**

votes

**2**answers

202 views

### Baire Category Theorem for complete uniform spaces

The version of the Baire Category Theorem I have in mind is the statement that a countable intersection of dense open subsets of a complete metric space is dense. The question is: is it likewise ...

**5**

votes

**2**answers

606 views

### Is there a topological Chevalley-Shephard-Todd Theorem?

Is the following true:
For a representation of a finite group $G$ on $\mathbb{C}^n$, the quotient $\mathbb{C}^n/G$ is a topological manifold if and only if $G$ is generated by pseudo-reflections.
( ...

**1**

vote

**0**answers

100 views

### Reference request: The compactness and compact embedding in Besov Space?

Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $0<s<1$, $1\leq p<\infty$, and $1\leq \theta\leq\infty$. We denote by $B^{s,p,\theta}(\Omega)$ the Besov space. For ...

**0**

votes

**0**answers

87 views

### A question about tensor product [on hold]

For every $f, g$ in $L^1(G)$, we know the function $[(x,y)↦f(x)g(y)]$ belongs to $L^1(G\times G)$ where here $\times$ means cartesian product. Why are these functions dense in $L^1(G\times G)$?

**4**

votes

**4**answers

220 views

### When is the boundary of an open planar set a Jordan curve?

Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve?
Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem.
My ...

**5**

votes

**2**answers

169 views

### If $f : [-a,a] \rightarrow \mathbb{IR}$ is Scott continuous, why are $f^-$ and $f^+$ measurable?

In "A domain theoretic account of Picard's theorem" (http://www.doc.ic.ac.uk/~dirk/Publications/icalp2004.pdf), the authors assert the following.
Let $\mathbb{IR}$ be the interval domain $\lbrace ...

**1**

vote

**0**answers

72 views

### 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 ...

**5**

votes

**1**answer

152 views

### Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.
Is there a non-separable complex manifold? Can a non-separable complex ...

**2**

votes

**1**answer

205 views

### Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem.
Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...

**4**

votes

**2**answers

312 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**33**

votes

**3**answers

8k views

### Properly Discontinuous Action

When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...

**8**

votes

**2**answers

291 views

### Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...

**2**

votes

**0**answers

45 views

### Are all locally compact anisotropic groupoids etale up to equivalence?

By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ...

**0**

votes

**0**answers

110 views

### Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be ...

**4**

votes

**3**answers

384 views

### “Abnormal” manifold

We often assume manifolds to be paracompact Hausdorff. Clearly, this implies normal.
However, there is a manifold (I mean locally Euclidean Hausdorff space) which is not paracompact. Without ...

**7**

votes

**5**answers

2k views

### Is the long line paracompact?

A manifold is usually defined as a second-countable hausdorff topological space which is locally homeomorphic to Rn. My understanding is that the reason "second-countable" is part of the definition is ...

**5**

votes

**1**answer

506 views

### Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furthermore let $$ \mathcal{A} := \{ f \in ...

**-1**

votes

**0**answers

77 views

### Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?).
What ...

**2**

votes

**1**answer

92 views

### Exponential locales and a pointless version of the compact-open topology?

TL;DR: compact-open topology for Homs of locales?
Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales.
For two locales, $A$ and $B$, is there a nice way to make an ...

**1**

vote

**2**answers

76 views

### Is an open subset of a compact subset of a Hausdorff locally convex TVS paracompact?

This repeats the title in a more readable way. Take a compact subset $X$ of a Hausdorff locally convex topological vector space and $U$ be an open subset of $X$. Is $U$ paracompact?

**1**

vote

**0**answers

168 views

### Existence of topology on the space of continuous functions

Let $C:=C([0,1],\mathbb{R})$ be the space of real-valued continuous functions defined on $[0,1]$. Could we find a topological vector space topology $\pi$ on $C$ such that the following two conditions ...

**6**

votes

**1**answer

165 views

### What is $Sq^i(\alpha^j)$ for all $i$ and $j$?

Write $H^*(\mathbb{R}P^\infty; \mathbb{Z}_2) = \mathbb{Z}_2[\alpha]$, $\deg \alpha = 1$. What is $Sq^i(\alpha^j)$ for all $i$ and $j$? I am not an algebraic topologist by trade but need to know this ...

**1**

vote

**0**answers

11 views

### 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 ...

**5**

votes

**2**answers

185 views

### Does a continuous map $f$ from the $n$-ball $B$ into $R^n$ such that $B\subset f(B)$ have a fixed point?

If $f$ is a continuous map from the $n$-ball $B$ into itself, the Brouwer fixed point theorem guarantees a fixed point. What if we assume that $f$ maps $B$ into all $R^n$, and $f(B)$ contains $B$? For ...

**30**

votes

**3**answers

1k views

### Is there a subset of the plane that meets every line in two open intervals?

Using the Axiom of Choice, it is possible to construct a subset of the plane that meets every line in two points (these are called "$2$-point sets"). What if, instead of points, we ask for two open ...

**6**

votes

**1**answer

129 views

### (A kind of) Irreducibiliy of regular open convex sets in the Cartesian space

I am looking for a proof of the fact which is formulated at the bottom of this post. The property of regular convex sets which the fact expresses seems to be true to me, yet I have not been able to ...

**10**

votes

**2**answers

180 views

### Concrete examples of covering from the 3-torus to the 3-sphere

There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian ...

**7**

votes

**1**answer

201 views

### Does “$\forall Z(C(X,Z) \cong C(Y,Z))$” imply $X\cong Y$?

If $X, Y$ are topological spaces, let $C(X,Y)$ denote the collection of continuous maps $f: X\to Y$, endowed with the compact-open topology.
Assume that we are given topological spaces $X,Y$ such ...

**2**

votes

**1**answer

82 views

### A quasicompact space with a net that contains no convergent strict subnet

If $x:\Lambda \rightarrow X$ is a net in a topological space $X$ and $\Lambda '\subseteq \Lambda$ is a cofinal subset of the directed set $\Lambda$, then $x|_{\Lambda '}$ is a subnet of $x$. We call ...

**4**

votes

**0**answers

139 views

### Thom Class of tensor bundles

Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...

**8**

votes

**2**answers

207 views

### Coarsest admissible topology on $\text{Cont}(X,Y)$

Let $X, Y$ be topological spaces and let $\text{Cont}(X,Y)$ be the collection of continuous functions $f:X\to Y$. We say that a topology $\tau$ on $\text{Cont}(X,Y)$ is admissible if the evaluation ...

**2**

votes

**1**answer

311 views

### A question in general topology

Motivated by this question we ask:
Up to homeomorphism, are there only a finite number of connected locally compact hausdorff topological space $X$ such that $X$ has an open set $U$ ...

**1**

vote

**1**answer

264 views

### Weak topology on a pre-Hilbert Space

Since there was essentially no answers on my previous question, I will ask a partial case of it, which is very easy to state.
Let $\left(X,\left<\cdot,\cdot\right>\right)$ be an inner product ...

**1**

vote

**1**answer

289 views

### Weaker form of irreducible surjections

An irreducible surjection is usually defined as a continuous closed surjective map $f:X\rightarrow Y$ such that if for some closed set $C\subset X$ one has $f(C)=Y$ then $C=X$. In my dissertation I ...

**3**

votes

**2**answers

90 views

### Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$.
Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$.
However, can all these paths ...

**2**

votes

**1**answer

38 views

### Compact $R_1$-spaces

A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$.
If $X$ is compact and $R_1$, does ...

**1**

vote

**0**answers

33 views

### 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 ...

**13**

votes

**1**answer

396 views

### Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper
[Brian, Mislove, Every compact group can have a non-measurable subgroup].
A positive solution to a variation of the following problem implies a ...

**3**

votes

**1**answer

180 views

### Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.
What is minimum ...

**4**

votes

**2**answers

201 views

### Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

A set $X\subset \mathbb{R}$ is called nice if for every $\epsilon > 0$ there are a
positive integer $k$ and $k$ bounded intervals $I_1,I_2,...,I_k$ such that
$X \subset I_1 \cup I_2 \cup ...

**1**

vote

**1**answer

74 views

### Problem about the existence of a continuous surjective map [closed]

Let $F$ be a closed set in $\Bbb R^2$, $F\neq \varnothing,\Bbb R^2$, and $F^\circ\neq \varnothing$,
does there exist a continuous surjective map from $\Bbb R\times \Bbb Z$ to $F^{\circ-}-F^\circ$？

**2**

votes

**1**answer

45 views

### If $X$ has the “discrete” covering property, how about $X^2$?

We say that a space $X$ has covering property (C) if the following holds:
(C) For any open cover ${\cal U}$ of $X$ there is a closed discrete set $D\subseteq X$ and a map $\varphi: D\to {\cal U}$ ...

**3**

votes

**1**answer

57 views

### Example of a collection of metacompact spaces with non-metacompact box-product

Is there an example of a family $(X_i)_{i\in I}$ of metacompact spaces, such that their box product $\prod_{i\in I}^{\textrm{Box}}X_i$ is not metacompact?