**2**

votes

**3**answers

202 views

### If $X$ is compact, is $[X]^2$ compact, too?

Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in ...

**0**

votes

**0**answers

118 views

### How to give a $\Delta$-complex structure?

The quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic with $\mathbb{R^2}$.
But I am not able to prove , ...

**6**

votes

**3**answers

283 views

### Hausdorff space $X$ with $X\cong [X]^2$

Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in ...

**5**

votes

**1**answer

118 views

### Example of a $G$-sphere that is not a $G$-representation sphere

Let $G$ be a finite group with the discrete topology. To set terminology:
a $G$-sphere is a sphere equipped with a continuous $G$-action
a $G$-representation sphere is a $G$-sphere obtained from an ...

**4**

votes

**2**answers

635 views

### Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish?
I have a specific space in mind, so if the ...

**2**

votes

**1**answer

50 views

### Intersection of complements of connected components (2)

Let $(X,d)$ be a non-compact, complete metric space and $K\subseteq X$ compact. Pick $x^* \in X\setminus K$.
Let $E$ be the connected component of $X\setminus K$ that contains $x^*$. Let ${\cal C}$ ...

**4**

votes

**1**answer

29 views

### Intersection of complements of connected components

Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^*\in X\setminus S$.
Let $E$ be the connected component of $X\setminus S$ that contains $x^*$. Let ${\cal C}$ be the ...

**5**

votes

**2**answers

173 views

### Closure of the graph of a function

Let $X_1, X_2$ be non-empty sets and let $R\subseteq X_1\times X_2$ such that for all $x\in X_1$ there is $y\in X_2$ such that $(x,y)\in R$.
Are there topologies $\tau_i$ on $X_i$ for $i=1,2$ and a ...

**9**

votes

**1**answer

230 views

### Meager subgroups of compact groups

Suppose we have an infinite compact (Hausdorff) group $G$, and a subgroup $H\leq G$ which is meagre.
Can $H$ always be covered by a countable family of nowhere dense sets $H_n$ such that $H_n^2$ is ...

**0**

votes

**0**answers

66 views

### Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?

**0**

votes

**1**answer

112 views

### Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all:
Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...

**3**

votes

**0**answers

108 views

### Inverse limit in shape theory

Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...

**4**

votes

**1**answer

99 views

### Measurability and continuity for general topological spaces

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K ...

**3**

votes

**1**answer

219 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, ...

**4**

votes

**0**answers

67 views

### Idempotent relations on the unit square with closed graphs

A colleague and I are interested in idempotent relations from $I=[0,1]$ to $I$ - relations such that $R\circ R(x)=R(x)$ for all $x\in I$. Specifically, the graphs of the relations we care about must ...

**4**

votes

**2**answers

295 views

### The classifying space of an infinite totally ordered set is contractible

I asked this question on math.stackexchange, but no one answered.
Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...

**5**

votes

**1**answer

114 views

### TVS with null topological dual space

In that post, I give an example of a TVS for which the topological dual is equal to $0$. But in the example, there is no open convex subset different from the empty set or the space itself.
Do you ...

**3**

votes

**2**answers

171 views

### Examples of TVS with no non-trivial open convex subsets

I give here the classical example of the space $E = L^p([0,1])$ which has no open convex subsets apart from $\emptyset$ and $E$. Consequently, there is no non-trivial continuous linear form on $E$.
...

**-3**

votes

**1**answer

183 views

### Loop space of manifold [closed]

Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...

**3**

votes

**2**answers

296 views

### Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements.
Let $h,k\geq1$ be integer numbers and let ...

**29**

votes

**0**answers

759 views

### Bidi: A new cardinal characteristic of the continuum?

This question assumes familiarity with combinatorial cardinal
characteristics of the continuum.
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural ...

**5**

votes

**0**answers

123 views

### The dominating number $\mathfrak{d}$ and convergent sequences

All spaces considered below are compact Hausdorff.
If $K$ is a space, then $w(K)$ is its weight. For a Boolean algebra $\mathcal{A}$, $K_\mathcal{A}$ denotes its Stone space. I am interested in ...

**0**

votes

**1**answer

154 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 ...

**5**

votes

**1**answer

111 views

### In the category of uniform spaces, is the completion of a quotient map also a quotient map?

I asked this question about 2 months ago on math.stackexchange, but so far I received neither comments nor answers.
Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous ...

**4**

votes

**0**answers

89 views

### 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 ...

**5**

votes

**1**answer

357 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 ...

**4**

votes

**1**answer

166 views

### Uncountable intersections of open balls in a separable metric space

Let X be a separable metric space, possibly assumed to be complete, and $B_i, i \in J$ an infinite collection of open balls. Is it true that there always exists a countable subset K of J such that the ...

**0**

votes

**1**answer

75 views

### Connectedness of the complements of the connected subsets

EDIT: My original foolish version was instantly destroyed by Dylan Thurston; it consisted of questions 1 & 2 below. Thus now only new question 0 remains to be answered.
Let $\ X:=M^n\ $ be a ...

**8**

votes

**1**answer

202 views

### Contractible and Delta-generated implies strong deformation retract to a point?

If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point.
However for general spaces it is well-known that just because a space is contractible, it does ...

**2**

votes

**1**answer

114 views

### Zero-dimensional spaces and clopen separations

Let $X$ be a topological space. (All of the spaces I'm considering are $T_0$, but in general they are not $T_1$. To be even more concrete, one can even consider $X={\rm Spec}(R)$ to be the space of ...

**1**

vote

**1**answer

124 views

### Union of connected sets

$\forall \beta \in I$, $A_{\beta }$ is connected, and $\left ( \bigcup_{\alpha < \beta }A_{\alpha } \right )\cap A_{\beta }\neq \varnothing$ . Is $\bigcup_{\alpha \in I}A_{\alpha } $connected?
For ...

**-1**

votes

**1**answer

111 views

### Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...

**2**

votes

**1**answer

150 views

### Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus.
Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...

**2**

votes

**1**answer

91 views

### Priestley topologizability and connected components

This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley ...

**12**

votes

**1**answer

621 views

### What if homotopy were expanded to allow any connected space instead of [0,1]?

What would happen to homotopy theory if we used a more general definition of homotopy, based on general connected spaces rather than [0,1]?
Given continuous $f,g:X\to Y$, define $f$ and $g$ to be ...

**4**

votes

**1**answer

190 views

### The subbase theorem for total boundedness

The uniform space analogue of Alexander's subbase lemma on compact subbase is (As we know, Alexander subbase lemma can be used to prove Tychonoff's theorem) :
Let $(X,\mathcal{U})$ be a uniform ...

**4**

votes

**1**answer

145 views

### convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on
a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$.
Suppose I know that
$$\int f d \mu_n \to \int f d\mu$$
...

**5**

votes

**2**answers

181 views

### Extending hyperconnected spaces

A hyperconnected space is a topological space such that every two non-empty open sets have non-empty intersection. Let's call a space $(X,\cal{T})$ maximally hyperconnected if it is hyperconnected and ...

**21**

votes

**2**answers

1k views

### Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...

**5**

votes

**2**answers

110 views

### Minimal Hausdorffness reversed

It turns out that not every Hausdorff topology is contained in a minimal Hausdorff topology. Let's put this question on its head: is every non-$T_2$ topology contained in a topology that is maximal ...

**1**

vote

**0**answers

86 views

### Two questions on hyperspace of a metric space

Let $(X,T)$ be a compact metrizable space. For every metric $d$ on $X$ which is $T-$ compatible, the Hausdorff metric on $2^{X}$ gives a topology on $2^{X}$.
(Up to homeomorphism) is this topology ...

**3**

votes

**0**answers

60 views

### Is the order convergence topology on a poset always Hausdorff?

In this post two topologies on a poset $(P,\leq)$ were defined: the interval topology $\tau_i(P)$ and the order convergence topology $\tau_o(P)$. It turns out that $\tau_i(P)$ is always $T_1$ and ...

**6**

votes

**2**answers

195 views

### Does every locally compact Hausdorff space admit a locally finite open covering by relatively compact sets?

Let $X$ be a locally compact Hausdorff space. Does there exist a locally finite open covering consisting of relatively compact sets?

**2**

votes

**1**answer

90 views

### Interval topology and order convergence topology

Throughout this post, 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 ...

**2**

votes

**1**answer

136 views

### Mean on compact metric spaces

Let $X$ be a compact metric space. A $k$ mean on $X$ is a continuous map $f:X^{k}\to X$ which is identity on the diagonal and is invariant under all $k$-permutations. For details, See the following ...

**4**

votes

**2**answers

191 views

### Two notions of zero-dimensionality for topological spaces

Let $(X,\tau)$ be a topological space.
We say that $(X,\tau)$ is zero-dimensional with respect to the Lebesgue covering dimension (zd1) if every open cover of the space has a refinement which is a ...

**-1**

votes

**4**answers

520 views

### Studying topology: which first, algebraic or differential? [closed]

I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...

**5**

votes

**1**answer

148 views

### Are all minimal totally separated spaces compact?

Let us call a space $(X,\tau)$ totally separated if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with $\sigma\neq ...

**4**

votes

**1**answer

148 views

### Minimal totally separated spaces

Let us call a space $(X,\tau)$ totally separated (ts) if for every two distinct points there is a clopen set containing one, but not the other. If for every topology $\sigma\subseteq\tau$ with ...

**3**

votes

**2**answers

101 views

### Connected, maximal compact, but not $T_2$

Is there a connected topological space that is maximal compact, but not $T_2$? (A space $(X,\tau)$ is said to be maximal compact if for any topology $\tau'$ on $X$ with $\tau'\supseteq \tau$ and ...