**0**

votes

**0**answers

58 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 ...

**2**

votes

**1**answer

304 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 $U$ ...

**2**

votes

**2**answers

98 views

### 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 bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are ...

**1**

vote

**1**answer

262 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

288 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

71 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

34 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

27 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 ...

**12**

votes

**1**answer

376 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

168 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

191 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

73 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 $F^{\circ-}-F^\circ$？

**-1**

votes

**0**answers

86 views

### Graph Theory text for a social scientist [closed]

I am a graduate student in Economics. I have a decent grounding in maths, but I've never studied graph theory or combinatorics. I need to study graph theory in order to analyse production networks. ...

**2**

votes

**1**answer

43 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

54 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?

**3**

votes

**1**answer

62 views

### Can any $n$ dimensional (smooth, PL, topological) closed manifold be covered by $2^n$ pieces of $n$ dimensional real spaces?

For any $n$ dimensional closed manifold $M^n$, can we find an open covering $\{U_i\}_{i\in[2^n]}$ such that $M=\cup U_i$ and each $U_i\cong \mathbb R^n$? How about complex manifolds (replacing ...

**-1**

votes

**0**answers

90 views

### Embedding uncountably many pseudo-circles in a closed annulus

Following up on my previous question about Cantor sets:
Is it possible to embed uncountably many pseudo-circles in a closed annulus so that any two are disjoint and so that each of them separates ...

**2**

votes

**1**answer

195 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 ...

**0**

votes

**0**answers

183 views

### 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 ...

**4**

votes

**2**answers

181 views

### Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

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 ...

**3**

votes

**0**answers

72 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 $\bigcup {\cal C} = X$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set, we define the ...

**0**

votes

**0**answers

57 views

### 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 ...

**2**

votes

**1**answer

169 views

### Embedding uncountably many disjoint copies of the Cantor set in the interval [closed]

Is it possible to embed uncountably many copies of the Cantor set in the unit interval so that any two are disjoint?

**5**

votes

**3**answers

193 views

### Which topological properties are preserved under taking box products?

Although the box topology is a topology worth studying and is similar to the strong topology in differential topology, the box topology is in many regards very badly behaved since the box product of ...

**1**

vote

**0**answers

195 views

### On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix.
Let $J$ be all $1$ matrix.
Let $\bar{A}=J-A$.
Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...

**2**

votes

**1**answer

204 views

### A homeomorphism between total spaces with same fiber and base spaces not homotopic

Is there a counterexample to the following assertion?:
Let $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ be fibrations with the same fiber $\mathbb S ^1$ such that $E_1$ and $E_2$ are homeomorphic (and both ...

**1**

vote

**1**answer

175 views

### Class of functions between $C^{\infty}$ and $C^{\omega}$

I am always curious about that whether there exists a class of function which seems that more smooth than the $C^{\infty}$ class, while it is far from $C^{\omega}$ analytic function .
From my point ...

**17**

votes

**4**answers

2k views

### Compact open topology on $\mathrm{Homeo}(X)$

Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. ...

**4**

votes

**0**answers

104 views

### Is there a universal $\omega$-limit set?

For the purposes of this question, a dynamical system means a compact metric space $X$ together with a continuous map $f: X \to X$.
For $x \in X$, the $\omega$-limit set of $x$, denoted $\omega(x)$, ...

**4**

votes

**0**answers

103 views

### Space of continuous real-valued functions on $[0,1]^\omega$ with the weak and pointwise topology

Let $C([0,1]^\omega)$ denote the set of continuous functions $f:[0,1]^\omega \to \mathbb{R}$. We endow $C([0,1]^\omega)$ with two topologies. Let $\tau$ be the pointwise topology on $C([0,1]^\omega)$ ...

**3**

votes

**0**answers

84 views

### $T_2$-space $X$ with $X\cong \text{Aut}(X)$

Is there an infinite $T_2$-space $X$ with $X\cong \text{Aut}(X)$? (Here, $\text{Aut}(X)$ is the set of automorphisms $\varphi:X\to X$ and it carries the topology inherited from the product topology on ...

**0**

votes

**2**answers

145 views

### Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?

Suppose $(G,\mathcal T)$ is a paratopological group and $a,b\in G$ and every neighborhood of $a$ contains $b$. Can we say every neighborhood of $b$ contains $a$?
clearly every closed neighborhood ...

**3**

votes

**1**answer

167 views

### Does the CGWH-fication change the (weak) homotopy type?

Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces.
There is the CG-ification ...

**4**

votes

**2**answers

167 views

### Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology.
The evaluation map $$ev\colon ...

**3**

votes

**0**answers

57 views

### Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer:
Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...

**-1**

votes

**0**answers

32 views

### Properties of Pushout [migrated]

suppose we have a pushout square in $\mathrm{Top}$:
\begin{align*}
\require{AMScd}
\begin{CD}
X_0 @>{\mu_1}>> X_1\\
@V{\mu_2}VV @VV{\alpha_1}V \\
X_2 @>>{\alpha_2}> X
...

**0**

votes

**1**answer

92 views

### Borel subsets of Polish groups

Suppose that I have a polish group $G$ and two subsets $A$ and $B$ of $G$ such that: $A$ is open in $G$ and $B$ is closed in $G,$ from this, can I conclude that $AB$ is a Borel subset of $G$? if not, ...

**14**

votes

**9**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...

**9**

votes

**0**answers

118 views

### A compact T1 topological space has a proper dense subset to which it is homeomorphic. What can be said about the space?

Let $X$ be a compact T1 (so singleton subsets are closed) topological space. Suppose that there is a proper subset $D \subset X$ such that:
$D$ is dense in $X$;
$D$ is homeomorphic to $X$.
Note ...

**11**

votes

**1**answer

295 views

### Non meager rectangle

Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?

**5**

votes

**1**answer

345 views

### Intersections of open balls in manifolds

This question is motivated by the post Uncountable intersections of open balls in a separable metric space.
The general problem is the following: given a connected Riemannian manifold $M$, what are ...

**6**

votes

**2**answers

227 views

### Space $X$ such that $X^\lambda\cong X$ for some $\lambda$

Which cardinals $\lambda > 2$ have the following property?
There is a space $(X,\tau)$ such that
for all cardinals $\kappa$ with $1<\kappa<\lambda$ we have $X\not\cong X^\kappa$, and
...

**3**

votes

**1**answer

209 views

### Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, ...

**0**

votes

**1**answer

162 views

### A question about open subsets of Hilbert space whose complements are compact sets

Let $H$ be an infinite-dimensional separable Hilbert space. Let $C$ be the intersection of a denumerably infinite sequence of sets, each of which is the complement of a compact subset of $H$. In other ...

**6**

votes

**2**answers

210 views

### Spaces that can't be embedded in the plane

If $X$ and $Y$ are topological spaces, let us write $X \preceq Y$ whenever $X$ embeds in $Y$.
Earlier today, I asked the question:
Is this a well-quasi-order on the completely metrizable spaces?
...

**2**

votes

**0**answers

95 views

### Normed space that is sigma-totally-bounded but is not sigma-compact

Q1: Is there a separable normed space that is not sigma-compact, but is a countable union of
totally bounded closed subsets?
A test case is the space $C^1(I)$ with the $C^0$ norm where ...

**-2**

votes

**1**answer

81 views

### Can every non-discrete topological group G be algebraically generated by a nowhere dense subset?

Is there somone help me to show that if this problem have positive Answer :
Problem :Can every non-discrete topological group G be algebraically gen-
erated by a nowhere dense subset ?
Thank ...

**0**

votes

**1**answer

151 views

### Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?

I am reading a blog which talks about a $C^1$ diffeomorphism $f: \mathbb{D}\{ x^2 + y^2 < 1\} \to \mathbb{R}^2$ and estimates the Hausdorff dimension of its image $\mathcal{H}_\sqrt{2}^d ...

**9**

votes

**1**answer

329 views

### Ultralimit versus partial limit

Let $\omega$ be a nonprincipal ultrafilter on $\mathbb N$.
A standard construction gives an $\omega$-limit, say $x_\omega$, for any bounded sequence $(x_n)$ of real numbers.
Namely, there is unique ...

**5**

votes

**0**answers

112 views

### Homogeneous $\omega$-monolithic compact space

Under CH, is the cardinality of every homogeneous $\omega$-monolithic compact space
$X$ not greater than $2^{\omega}$?