**4**

votes

**1**answer

258 views

### Free action of $\mathbb{Z}(2^{\infty})$ on a compact space

Assume that $X$ is a Hausdorff compact space such that $\forall n\in \mathbb{N}$, we have a free action of $\mathbb{Z/{2^{n}}\mathbb{Z}}$ on $X$. Must $\mathbb{Z}(2^{\infty})$ act freely on ...

**-4**

votes

**0**answers

36 views

### An exercise related to Arzela- Ascoli theorem [on hold]

I am looking for some hint to prove the set A={x belonging to C[a,b]| sup|x| +sup|x'|<=1} is compact. Can you help me? I will appreciate any help!

**0**

votes

**0**answers

104 views

### A question about a subset in R^n homeomorphic to an open subset [on hold]

Let A be a subset of n-dimensional Eucliean space R^n,
A is homeomorphic to an open subset of R^n.
Then whether A is also an open subset of R^n?
Is it a theorem in somewhere?
Thank you very much.

**0**

votes

**0**answers

56 views

### Characterizing subgroups $H$ of $\Bbb T$ with $\{(f(x),g(x))\mid x\in H\}$ dense in $\Bbb T^2$

Let $\Bbb T$ be the circle group with Euclidean topology. Is there a way to determine all $H\le \Bbb T$ such that there are $f,g\in Aut(H)$ with $\{(f(x),g(x))\mid x\in H\}$ dense in $\Bbb T\times ...

**2**

votes

**2**answers

127 views

### Is there an (almost) dense set of quadratic polynomials which is not in the interior of the Mandelbrot set?

For the parameter plane of complex quadratic polynomials, $(z\mapsto z^2+c)_{c\in\mathbb{C}}$ :
Is it possible to find a part of the parameter plane, scanned with a given limited precision ...

**4**

votes

**1**answer

145 views

### When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?

Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...

**0**

votes

**0**answers

28 views

### About Hausdorff characterization [closed]

I am thinking about why only in complete metric space a set A is compact if and only if A is totally bounded and closed? Anyone can help me? Thanks a lot!

**5**

votes

**1**answer

178 views

### “Productively normal” space

If a set $S$ is endowed with the discrete topology $\mathcal{P}(S)$, then for every normal space $N$ the product $S\times N$ is normal.
Question: can we endow a set $S$ with another Hausdorff ...

**-2**

votes

**0**answers

51 views

### Map between manifolds [closed]

Let $M,N \subset \mathbb{R}^3$ be (not necessarily smooth) 2-manifolds without boundary. Let $f: M \rightarrow N$ be a continuous function and suppose that $f$ is injective. Let $x \in M$ and let $U$ ...

**11**

votes

**1**answer

212 views

### Universal maps between topological spaces

Let $X,Y$ be topological spaces. We call a continuous map $u:X\to Y$ universal if for every continous map $f:X\to Y$ there is $x\in X$ such that $f(x) = u(x)$.
If $u:X\to Y$ and $v:Y\to Z$ are ...

**-1**

votes

**0**answers

56 views

### Decomposition of separable metric space with certain topological property [closed]

Is there any information about when separable metric space with certain topological property can be decomposed into finetely many zero-dimensional subspaces with the same property?
I am mainly ...

**1**

vote

**0**answers

81 views

### continuity with respect to weak-${\ast}$ topology

Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...

**2**

votes

**1**answer

60 views

### Constructivity of zeros demanded by topological degree

Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...

**3**

votes

**1**answer

58 views

### Characterizing space that preserves positive-definiteness property

Given a symmetric positive-definite matrix $\Sigma$, consider the space $\mathcal{D}$ of diagonal matrices such that $\forall D\in\mathcal{D}$, the matrix $\Sigma-D\Sigma^{-1}D$ is positive definite. ...

**3**

votes

**1**answer

98 views

### A question about Skorokhod metric

I have a question related to the Skorokhod distance.
Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...

**3**

votes

**1**answer

70 views

### Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$

Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g.
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
Now define ...

**1**

vote

**0**answers

103 views

### (The Homotopy type of the) lifting of homeomorphism of Grassmanian

For $k<n$ put $FM_{k\times n}$ for the space of all $k\times n$ full rank matrices with real or complex entries. Note that the permutation group $S_{n}$ has an obvious action on this space ...

**4**

votes

**1**answer

80 views

### Approximation of sets by sets with regular border

What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...

**5**

votes

**0**answers

168 views

### Gorelic's Forcing for large Lindelöf spaces with points $G_\delta$

I am trying to understand a step for proving that there exists large Hausdorff Lindelöf Spaces with points $G_\delta$ using forcing. I am following Isaac Gorelic's "The Baire Category And Forcing ...

**4**

votes

**0**answers

63 views

### Large discrete subspaces in spaces of separately continuous functions

For topological spaces $X,Y,Z$ let $SC_p(X\times Y,Z)$ be the space of separately continuous functions $f:X\times Y\to Z$ endowed with the topology of pointwise convergence.
It is easy to see that ...

**-1**

votes

**1**answer

197 views

### Creating topological spaces with portals [closed]

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...

**0**

votes

**0**answers

41 views

### Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...

**5**

votes

**1**answer

150 views

### When is the topology generated by countable subsets?

Let $X$ be a topological (Hausdorff) space and let $(X_\alpha)_\alpha$ be a directed family of subsets. We say that $(X_\alpha)_\alpha$ generates the topology of $X$ if a subset $U \subseteq X$ is ...

**3**

votes

**0**answers

81 views

### Order dimension vs topological dimension of a poset

Let $(P,\leq)$ be a partially ordered set (poset). We define the ordering dimension $\textrm{dim}_\textrm{ord}(P)$ of $(P,\leq)$ to be the smallest cardinal $\kappa$ such that there exist a set of ...

**-1**

votes

**1**answer

53 views

### extension of a continuous function [closed]

Please is it true that if $f:K\to \mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of $K$?
...

**-1**

votes

**0**answers

65 views

### complete compact open topology [migrated]

Let $X$ denotes a path-connected and compact manifold and $PX$ its path-space (the set of continuous maps $\gamma: [0,1] \longrightarrow X$) topologized with the compact open topology. It is true that ...

**4**

votes

**1**answer

192 views

### If $S\subset\mathbb R$ is a $G_\delta$ there is a function $\mathbb R\to\mathbb R$ continuous exactly on $S$. Reference?

Let $S\subset\mathbb R$ be a $G_\delta$ set. A variation on the construction of the Thomae function (which is discontinuous on the rationals and continuous elsewhere) shows that there is a function ...

**4**

votes

**0**answers

114 views

### Examples of topologies compatible with a given dual pair

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...

**6**

votes

**0**answers

116 views

### A forked plane continuum

I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...

**0**

votes

**1**answer

49 views

### Existence of half-planes with respect to regular open sets of the Euclidean plane

I initially asked this question at math.stackexchange.com but there was no reaction, so I thought this may be a good idea to transfer it to mathoverflow.net
Let ...

**4**

votes

**2**answers

242 views

### Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...

**1**

vote

**1**answer

82 views

### Name for (function, set) pairs?

Right now I'm working on a topological graph theory problem. To prove a theorem I introduced some objects. Has anyone heard of something similar before? I would like to call them by the right name.
...

**2**

votes

**1**answer

67 views

### Special retraction from a metric space onto an arc

Suppose $X$ is a metric space and $A$ is a subspace of $X$ homeomorphic to $[0,1]$ with its usual topology. Let $v$ an end point of $A$, that is $v$ does not disconnect $A$. Is there a retraction $r$ ...

**6**

votes

**1**answer

91 views

### Universality with respect to quotients

Is there an infinite cardinal $\kappa$ for which the following statement (S) true?
(S) : There is a topology $\tau_\kappa$ on $\kappa$ such that for all topological spaces $(X,\tau)$ with $|X|\leq ...

**2**

votes

**2**answers

204 views

### A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another
metric $d_{f}$ on $Y$ With $$ d_{f}(y_{1},y_{2})=Hd(f^{-1}(y_{1}), ...

**4**

votes

**0**answers

66 views

### The name for the quotient property

I asked this question on math@stackoverflow and was suggested to ask it here as well.
We call a surjective $f:X\rightarrow Y$ a quotient mapping if it satisfies, for every $U\subset Y$
...

**6**

votes

**1**answer

160 views

### cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$.
Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$.
Let $\rho: ...

**2**

votes

**0**answers

146 views

### A possible generalization of the Borsuk Ulam theorem via action of symmetric groups

The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...

**2**

votes

**1**answer

67 views

### Unbounded convex not containing a ray - example without using a basis

I prove here that an unbounded convex in a finite dimensional space contains a ray. At the same place, I give an example of an unbounded convex not containing a ray in the case of an infinite ...

**5**

votes

**1**answer

507 views

### A good book on adeles and ideles

Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, ...

**3**

votes

**2**answers

147 views

### Is every Montel locally convex vector space compactly generated?

Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that $X$ is a semi-Montel space if every ...

**2**

votes

**0**answers

99 views

### A topological space extracting from a group action

Let $G$ be a compact abelian topological group with invariant measure $\mu$ which acts on a compact Hausdorff space $X$. A $G$-odd function is a continuous function $f:X\to ...

**6**

votes

**0**answers

115 views

### Face poset of a subcomplex complement

Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\simeq$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that ...

**12**

votes

**1**answer

245 views

### Discrete subsets in the topology of pointwise convergence vs. metrisability

While reading Arkhangel'skii's Topological function spaces, I encountered an unexpected application of Martin's Axiom. This is Theorem II.5.20:
Assume $\mathsf{MA}+\neg \mathsf{CH}$. Let $X$ be a ...

**0**

votes

**0**answers

16 views

### Locally finite refinement with restricted class of covering sets

Let $X$ be a topological space, and let $U \subset X$ be paracompact.
This means that any open cover $(U_i)_{i \in I}$ of $U$ has a locally finite refinement.
In this refinement $(V_i)_{i \in I}$ of ...

**0**

votes

**1**answer

79 views

### Questions about the dimension-and other properties-of a non-separable topological space

Let the topological space X be the so-called "long line" — which is an uncountable linearly ordered set containing all the countable ordinal numbers as well as a copy of the open unit interval ...

**7**

votes

**2**answers

222 views

### Must a map on a compact space be surjective on $\cap_{n=1}^\infty f^n(X)$?

Maybe I'm just being a bit dense here, but this has me stumped right now.
A fairly well-know thm is the following: Let $X_0$ be a compact metric space and $f:X_0\to X_0$ be continuous. For each ...

**6**

votes

**2**answers

211 views

### Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...

**1**

vote

**0**answers

53 views

### Does real analytic imply locally contractible?

The statement is true for complex analytic spaces. I am not sure who proved this result.
I ask the same question in the real case.

**2**

votes

**1**answer

97 views

### Totally bounded group topologies on $\Bbb Q$ with trivial intersection of two neighborhoods

Are there totally bounded group topologies $\mathcal S$ and $\mathcal T$ on $\Bbb Q$ such that for some open sets $A\in\mathcal S$ and $B\in \mathcal T$ we have $A\cap B=\{0\}$?