Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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3
votes
1answer
169 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 ...
3
votes
0answers
63 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 $Y$ ...
0
votes
1answer
94 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, ...
9
votes
0answers
122 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 ...
4
votes
2answers
178 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 ...
11
votes
1answer
302 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 $A \times B \subseteq G$?
4
votes
2answers
204 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 ...
7
votes
2answers
236 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 ...
2
votes
0answers
100 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
1answer
83 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 ...
6
votes
2answers
216 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? ...
5
votes
0answers
115 views

Homogeneous $\omega$-monolithic compact space

Under CH, is the cardinality of every homogeneous $\omega$-monolithic compact space $X$ not greater than $2^{\omega}$?
11
votes
3answers
507 views

The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?
1
vote
0answers
72 views

Monadicity of profinite algebras

We can show that the category of profinite algebras, cofiltered limits of finite algebras, is monadic over Stone spaces as follows. So, I wonder if there are any other examples. In case that I was ...
11
votes
1answer
196 views

Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$ The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...
7
votes
0answers
129 views

Intersection of connected components in $\mathbb{R}^n$

Let $n$ be a positive integer and let $K\subseteq \mathbb{R}^n$ be compact. Pick $x^* \in \mathbb{R}^n\setminus K$. Let $E$ be the connected component of $\mathbb{R}^n\setminus K$ that contains ...
1
vote
1answer
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 $F^{\circ-}-F^\circ$?
5
votes
1answer
110 views

Does the property of being a local homeomorphism descend through split surjections?

Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps (between topological spaces). Assume these hypotheses: $f : X \to Y$ is a split surjection, i.e. has a section. $g \circ f : X \to Z$ is a ...
2
votes
3answers
201 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
0answers
114 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 , ...
5
votes
3answers
277 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
1answer
117 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
2answers
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 ...
1
vote
1answer
46 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 ${\cal C}$ ...
3
votes
1answer
27 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 ...
4
votes
2answers
169 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 ...
5
votes
0answers
85 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
0answers
64 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
1answer
107 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
0answers
103 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 ...
3
votes
1answer
97 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
1answer
213 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
0answers
65 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
2answers
292 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
1answer
111 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
2answers
170 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
1answer
178 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
2answers
292 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
0answers
748 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 ...
4
votes
0answers
120 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
1answer
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
1answer
106 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 ...
3
votes
0answers
87 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
1answer
351 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
1answer
162 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
1answer
74 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
1answer
196 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
1answer
112 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
1answer
122 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
1answer
109 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 ...