**5**

votes

**0**answers

107 views

### Uniform approximation of separately continuous functions on zero-dimensional spaces

For topological spaces $X,Y,Z$ а function $f:X\times Y\to Z$ is called separately continuous if for any $(x,y)\in X\times Y$ the restrictions of $f$ to the sets $\{x\}\times Y$ and $X\times \{y\}$ are ...

**9**

votes

**0**answers

160 views

### Embedding $\beta\mathbb{N}$ into a product of Cantor sets

Let us consider $\beta\mathbb{N}$, the Stone-Czech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...

**5**

votes

**0**answers

140 views

### On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...

**2**

votes

**0**answers

58 views

### Given locally compact and $\sigma$-compact, can we get partition of unity?

Let $X$ be a locally compact, $\sigma$-compact Polish (complete and separable metric) space. How to prove: "There is an increasing sequence of continuous cut-off functions with compact support, ...

**3**

votes

**1**answer

108 views

### Continuous and open image of a Polish space

Suppose that we have a continuous open and closed surjection $f\colon X\to Y$ of a Polish space $X$ to $Y.$ The closeness of $f$ implies that $Y$ is a metric space.
But i do not know how to use ...

**3**

votes

**2**answers

208 views

### Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?

In a paper that I am reading there is a following step:
Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$.
Then $\overline{co}(x_k)$ is a ...

**4**

votes

**0**answers

230 views

### Examples of a topological semidirect product

Let $G$ be a compact topological group, and $\operatorname{Aut}(G)$ the group of autohomeomorphisms of $G$. I have proved some (topological) results about the holomorph $G\leftthreetimes ...

**1**

vote

**2**answers

89 views

### Retract embedding of $S^{n}$ in its unit tangent bundle

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question:
For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?

**1**

vote

**0**answers

90 views

### Examples of value quantales

In his paper "Quantales and continuity spaces" R. C. Flagg gives the following examples of value quantales: the lattice $\bf{2}$ of truth values with usual addition, the lattice $\mathbb{R}_{+}$ of ...

**8**

votes

**1**answer

195 views

### Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...

**3**

votes

**0**answers

129 views

### Which Topological Spaces are Powers?

Given a topological space $X$ and closed subspace $Y \subset X$, it may be the case that $X$ is a power of $Y$. That means $\displaystyle X = \prod_{i < \kappa} Y_i$ for some cardinal $\kappa$ ...

**4**

votes

**5**answers

552 views

### Must uncountable compact Hausdorff spaces have large discrete subsets?

The situation is this. I have a space $X$ which is second countable, compact, and Hausdorff (it's a modified form of a type space, though I don't think that matters here). It has size continuum. It ...

**1**

vote

**0**answers

66 views

### Surjectivity of maps between spheres [closed]

I am wondering how to prove that a non-zero degree map from $S^n \to S^n$ is surjective. For example, identifying $S^1 \subset \mathbb{C}$, we can take $f:S^1 \to S^1$ via $f(z) = z^k$ with $k\neq 0$. ...

**4**

votes

**4**answers

474 views

### When is the quotient by an $n$-fold loop space an $m$-fold loop space?

Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about ...

**2**

votes

**2**answers

94 views

### Special filters in the algebra of regular open sets of a topological space

Let $\mathrm{r}\mathscr{O}$ be the family of open domains (regular open sets) of a topological space $\langle X,\mathscr{O}\rangle$, that is:
$$A\in\mathrm{r}\mathscr{O}\iff ...

**1**

vote

**1**answer

70 views

### Similarity graph for continuous maps between Hausdorff spaces

Let $X, Y$ be topological spaces and $f,g: X\to Y$ continuous. Then we say that $f, g$ are similar if for all $V\subseteq Y$ open we have either
$f^{-1}(V) = g^{-1}(V) = \emptyset$, or
$f^{-1}(V) ...

**-1**

votes

**1**answer

204 views

### An infinite set in a compact space

Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...

**0**

votes

**1**answer

60 views

### When is any convergence sequence is stationary?

Is there any characterization for a topological space under which every convergent sequence is stationary? (e.g. discrete topology)

**3**

votes

**0**answers

387 views

### Some counter examples in group theory

In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in ...

**2**

votes

**1**answer

116 views

### Intersection of compact sets in the compact-open topology

Let $(X,\tau)$ be a topological space. We topologize $\tau$ itself in the following way. For $K\subseteq X$ compact, we set $${\cal V}_K=\{U\in \tau: U \supseteq K\}.$$ The compact-open topology on ...

**6**

votes

**0**answers

110 views

### Cutting a piece of cake that $n$ people value as exactly $w$

Stromquist and Woodall (1985) study the problem of Sets on which several measures agree. There are $n$ non-atomic value measures on the unit circle, and a parameter $w\in(0,1)$. The goal is to find a ...

**5**

votes

**0**answers

113 views

### Is a successor to a successor to the trivial group topology on an Abelian group, totally bounded?

Is there an example of an Abelian group $G$ and group topologies $\cal S$ and $\cal T$ on it such that $\cal S$ is an immediate successor to the trivial topology on $G$ (i.e there is no other group ...

**4**

votes

**0**answers

57 views

### Does the $D$-property have universal objects?

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...

**1**

vote

**0**answers

88 views

### Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...

**3**

votes

**0**answers

51 views

### Adjunctions of uniformly locally connected spaces

A space $X$ is uniformly locally connected (ULC) if there exists an open neighbourhood $U$ of the diagonal $\vartriangle_X$ in $X \times X$ and a homotopy $H: U \times I \to X$ between $\pi_1|U$ and ...

**39**

votes

**1**answer

1k views

### Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?

Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...

**0**

votes

**1**answer

60 views

### Does order-preserving equal continuous? [closed]

Let $P,Q$ be posets and endow them with the interval topology $\tau_i(P)$ and $\tau_i(Q)$ respectively. Is it true that if $f: P\to Q$ is order-preserving, then it is continuous, and vice versa?

**1**

vote

**1**answer

68 views

### Path-connected Hausdorff interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected and $T_2$. Does this imply that $[0,1]$ order-embeds into $P$?
(This is a follow-up ...

**3**

votes

**1**answer

86 views

### Path-connected interval topologies

Let $(P,\leq)$ be a poset with more than $1$ point such that the interval topology $\tau_i(P)$ is path-connected. Does this imply that $[0,1]$ order-embeds into $P$?

**3**

votes

**1**answer

142 views

### Properties of the interval topology of the lattice of functions

Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ 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: ...

**12**

votes

**2**answers

387 views

### Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction.
Given diagrams of topological spaces
$$X_0\rightarrow X_1\rightarrow\ldots$$
$$Y_0\rightarrow Y_1\rightarrow\ldots$$
...

**3**

votes

**0**answers

97 views

### Cardinality based results in Topological Vector Spaces?

Given a topological vector space $V$, let its density be the smallest cardinal $A$ such that a set of cardinality $A$ is dense in $V$. Naively, it seems one of two things happen:
TVS's $V$ of ...

**1**

vote

**1**answer

106 views

### Quotients of powers of the Sierpinski space

Is every space isomorphic to some quotient of a power of the Sierpinski space?
More precisely: Let $(X,\tau)$ be a topological space, and let $\mathbb{S} = (\{0,1\}, \{\emptyset, \{0\},\{0,1\})$ be ...

**2**

votes

**0**answers

71 views

### Quotients of simplicial complexes which are simplicial complexes

In the category of topological spaces, I would like to know that quotients of simplicial complexes (or $\Delta$-complexes) by equivalence relations which are "unramified" in a suitable sense still ...

**4**

votes

**1**answer

119 views

### Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:
A topological dynamical system $(X, f)$ consists of a topological space (typically compact ...

**1**

vote

**0**answers

169 views

### Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions

I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to ...

**1**

vote

**0**answers

47 views

### 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 ...

**1**

vote

**0**answers

171 views

### 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 ...

**-3**

votes

**2**answers

124 views

### Corresponding between prime ideals in $C(X)$ and $C^*(X)$

we know that every maximal ideal in $C(X)$ is in this form:
$$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$
and every maximal ideal in $C^*(X)$ is
...

**0**

votes

**1**answer

51 views

### Rank of a generall linear group over a finite field [closed]

What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.

**14**

votes

**3**answers

449 views

### Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...

**2**

votes

**1**answer

124 views

### Reconstructing relations with the image relation of a topology

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R_{im}(X,\tau)$ is reflexive. This ...

**1**

vote

**1**answer

56 views

### Hausdorff spaces with asymmetric image relation

For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R_{im}(X,\tau)$ is reflexive, and ...

**5**

votes

**0**answers

167 views

### Topological Subset Take-Away

David Gale's subset take-away game is a game where two players A and B play with a finite set $S$. Players alternately choose proper nonempty subsets of $S$ such that if a subset is chosen, then none ...

**15**

votes

**1**answer

342 views

### The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...

**6**

votes

**3**answers

610 views

### Countable path-connected Hausdorff space

Is there a topology $\tau$ on $\omega$ such that $(\omega,\tau)$ is Hausdorff and path-connected?

**0**

votes

**1**answer

83 views

### “Universal” connected spaces

Let $\kappa$ be an infinite cardinal. Does there exist a topology $\tau_{\kappa+1}$ on $\kappa+1$ such that for any topological space $(X,\tau)$ with $|X|=\kappa$ the following statement is true?
...

**2**

votes

**1**answer

53 views

### Shrinkable decompositions with uncountably many non-degenerate elements?

Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...

**11**

votes

**2**answers

382 views

### Connected but no path connected components

Is there a Borel subset of plane which is connected but whose only path connected components are singletons?
I know that a Bernstein set is a non Borel example of such a set. Thanks!

**0**

votes

**0**answers

40 views

### Is the covering property $\Omega \choose \text{T}$ closed under products?

Suppose that $(X_i)_{i\in I}$ is a family satisfying the covering property $\Omega \choose \text{T}$ (for the definition of this covering property, see this post).
Does $\prod_{i\in I} X_i$ ...