**5**

votes

**1**answer

188 views

### Nonmetrizable compact totally disconnected spaces without isolated points

Brouwer proved that a topological space is homeomorphic to the Cantor set if and only if
1) it's non-empty,
2) it's compact,
3) it's totally disconnected,
4) it has no isolated points, and
5) ...

**3**

votes

**1**answer

208 views

### Continuous Functions

Is there a pair of continuous surjective functions say $f_1$ and $f_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f_1^{-1}(x) \cap f_2^{-1}(y)$ is non-empty?

**1**

vote

**0**answers

141 views

### The image of homomorphism of fundamental group of closed surface [closed]

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus >=2. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to S$ ...

**1**

vote

**3**answers

179 views

### Axiomatization of locally compact Hausdorff spaces via compact subspaces

The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets.
It ...

**1**

vote

**1**answer

75 views

### Simply connectedness of minimal resolution of Kleinian singularities

Is the minimal resolution of Kleinian singularities of type $D_k$ (i.e. the minimal resolution of singularities of the action of the binary dihedral group of order $4(k-2)$ on $ C^2$ simply connected? ...

**5**

votes

**1**answer

193 views

### If a subset and its complement are path-connected, an neighborhood of the subset is path-connected

I apologize if this is too elementary for this site.
Given a closed subset, $X\subset \mathbb{R}^n$, given $X$, $X^C$ path-connected, show that any path-connected neighborhood of $X$, denoted $M$, ...

**1**

vote

**1**answer

221 views

### (n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
...

**7**

votes

**3**answers

793 views

### Is the list of “known” 3D compact manifolds complete?

"it is an open question if the known compact manifolds in 3-D are complete."
This is a quote from Eric Weisstein's
CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480.
(Google ...

**4**

votes

**0**answers

112 views

### Pettis Integrability and Laws of Large Numbers

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...

**0**

votes

**0**answers

88 views

### non-Hausdorff paratopological group with closed intersection of neighborhoods of 1

Do you have an example of a non-Hausdorff (preferably locally compact) paratopological group $(G,\mathcal T)$ in which the intersection of all neighborhoods of a point is closed?

**0**

votes

**0**answers

57 views

### Is there a collectionwise normal topological vector space which is not paracompact?

I am looking for an example of a collectionwise normal topological vector space that is not paracompact. Any idea about it?

**13**

votes

**4**answers

482 views

### Continuity on a measure one set versus measure one set of points of continuity

In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...

**0**

votes

**0**answers

58 views

### How to construct point finite covering in collectionwise normal spaces

I am actually looking for a related reference (and ideally if anyone knows the answer) on the following construction problem:
Let X= $\prod_{i=1,..,n} X_{i}$ be a collectionwise normal and Hausdorff ...

**27**

votes

**2**answers

2k views

### Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function.
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e.
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...

**0**

votes

**0**answers

43 views

### tensors invariant under irregular flow

Suppose $M$ is a compact smooth manifold, and let $R$ be a nowhere-vanishing vector field. Then in the irregular scenario, the closure of $R$-orbits is a $r$-torus $T^r$.
Now suppose there is a ...

**0**

votes

**0**answers

116 views

### Subdividing Simplicial complexes

Barycentric subdivision is an important tool in simplicial homology theory, where it is used as a means of obtaining finer simplicial complexes from a given one.
Are there other common subdivision ...

**2**

votes

**1**answer

280 views

### Is this a closed set?

Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...

**1**

vote

**1**answer

89 views

### Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?

Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the ...

**2**

votes

**1**answer

227 views

### Dual of the space of continuous functions

Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by ...

**5**

votes

**0**answers

138 views

### Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...

**4**

votes

**1**answer

355 views

### Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...

**12**

votes

**0**answers

224 views

### Connected and locally connected, but not path-connected

Allow me to use some non-standard terminology:
A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to ...

**3**

votes

**2**answers

168 views

### Product of Topological Measure Spaces

Def. A Radon measure $\mu$ on a compact Hausdorff space $X$ is uniformly regular if there is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set ...

**1**

vote

**0**answers

136 views

### Extending a homeomorphism from a dense set [closed]

Let $X$ and $Y$ be Hausdorff topological spaces, and let $f : X \to Y$ be a Borel-measurable function. Suppose that $D \subseteq X$ is dense, that the image $f(D) \subseteq Y$ is dense, and that $f$ ...

**5**

votes

**4**answers

640 views

### Categorical Construction of Quotient Topology?

The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies ...

**12**

votes

**2**answers

257 views

### If G is a sequential topological group, must G x G be sequential?

Using standard definitions, the topological space $Y$ is sequential if for each nonclosed $A \subset Y$, there exists a convergent sequence $a_{1}$ , $a_{2}$,...$\rightarrow b$
so that $a_{n} \in A$ ...

**1**

vote

**0**answers

137 views

### Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define ...

**0**

votes

**0**answers

85 views

### Analytic extension from a closed analytic subset

I have the following question: Let $\Omega \subset \mathbb{R}^{n}$ be an open set and consider $X \subset \Omega$ an analytic subset. By this I mean that there exists analytic functions ...

**2**

votes

**2**answers

184 views

### Tychonoff spaces and ideals

Let $X$ be a tychonoff space and let $T$ be the set of all $f \in C(X)$ such that for any $g$ the equation $fg = 1$ has at most finitely many solutions. Under what conditions on $X$, the set $T$ is an ...

**5**

votes

**2**answers

478 views

### CW complex and group action

This is a general question and any reference or related result will be extremely helpful.
Suppose $X$ is a Hausdorff topological space. Suppose G (a countable group) acts on it. Let $Y=X/G$ be the ...

**2**

votes

**2**answers

202 views

### A Fixed point Theorem that does not need the convexity of set valued map?

I am looking for a fixed point theorem for set valued maps that does not assume the set valued map should be convex valued.
Something like contractiblity or other properties can be replaced with ...

**3**

votes

**1**answer

156 views

### Extending the hyperbolic splitting on $\Lambda$ to a neighborhood of $\Lambda$

Let $M$ be a compact Riemannian manifold and let $f:M→M$ a diffeomorphism. Let $\Lambda\subset M$ be a compact invariant subset of $M$. We say that $\Lambda $ is a hyperbolic set for $f$ when there ...

**2**

votes

**2**answers

164 views

### A set intersecting the graph of any continuous function in a finite set

New version of the problem I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true:
If $f :X\longrightarrow \Bbb{R}$ is continuous ...

**1**

vote

**1**answer

576 views

### Topological razors (ball-like spaces)

Introduction
Many admire the Euclidean space, and I am not an exception. I will try to catch the topological roundness of the $n$-ball in its greatest generality. I call the resulting axiomatized ...

**2**

votes

**1**answer

133 views

### Is there a maximal (or maximal Tychonoff) non normal space?

Is there a maximal (or maximal Tychonoff) non normal space? In "A Problem of Set-Teoretic Topology" the existence of a maximal Tychonoff space is asserted. Also there exists a perfectly normal maximal ...

**16**

votes

**2**answers

498 views

### Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...

**6**

votes

**2**answers

359 views

### “is topologically mixing” vs. “is topologically transitive” in the defition of chaos

This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.
Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"
as the ...

**3**

votes

**3**answers

311 views

### A question concerning the isomorphic type of continuous functions

let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it possible to consider $S$ as ...

**0**

votes

**2**answers

163 views

### Is there a normal space that is not uniformly normal

Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which
$$(\forall a\in A)(D[a]\subseteq B)$$
A ...

**1**

vote

**1**answer

80 views

### Generalized connected components decomposition for Priestley spaces

Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal ...

**0**

votes

**1**answer

218 views

### Which algebra of functions can be represented as $C(X)$

I don't know if this problem is known or not, so any information would be appreciated:
Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a ...

**1**

vote

**1**answer

83 views

### Normal Uniform Spaces and their function uniform spaces

Let $(X,\mathcal D)$ be a normal (diagonal) uniform space and $G$ be the set of all homeomorphisms $f:X\to X$. Let $\Delta$ be the uniformity on $X^X$ (inherited by $G$) by subbase
$$\Lambda =\{ ...

**2**

votes

**1**answer

86 views

### Are the connected components of a Priestley space closed?

Preliminaries A Priestley space is both a poset and a topological space. The topologically connected components of the space are trivially closed. (They are just the points of the underlying set.) But ...

**5**

votes

**1**answer

190 views

### What's the link between topological spaces as locales and topological spaces as infinity-groupoids?

I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...

**5**

votes

**1**answer

659 views

### A problem on infinite dimensional metric space

Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite ...

**1**

vote

**2**answers

155 views

### Metrization of spaces of functions

Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?
For $M$ compact and $N$ a metric ...

**4**

votes

**1**answer

226 views

### Omitting types and Baire category

What is the relation between omitting types theorems in model theory and the baire category theorem?

**4**

votes

**2**answers

293 views

### Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$

Let $X$ be the real line with the usual topology. Then clearly $|C(X)| = c = |X|$ and on the other hand $|X| = 2^{\aleph_0}$.
Now my question is as in the title: Is there a Tychonoff space $X$ of ...

**0**

votes

**1**answer

307 views

### What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here.
The Wikipedia article on convergence of measures defines three kinds of convergence: ...

**2**

votes

**1**answer

222 views

### A completely regular space that is very non-normal

Take a completely regular Hausdorff topological space $X$ considered as a subset of its Stone-Čech compactification $\beta X$. If $X$ is not normal, we can find a closed subset $Y$ of $X$ and a ...