**-1**

votes

**0**answers

10 views

### extension of a continuous function

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

178 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

100 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 ...

**0**

votes

**0**answers

37 views

### are these two groupoids Morita equivalent? [on hold]

We take the action groupoids $\mathbb{T} \rtimes_\alpha \mathbb{Z}_2$ and $\mathbb{T} \rtimes_\beta \mathbb{Z}_2$, where $\alpha(z)=-\bar{z}$ and $\beta(z)=\bar{z}$. Are these Morita equivalent?

**4**

votes

**0**answers

87 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

47 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

231 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

76 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

58 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

84 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

196 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

64 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

151 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

140 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

66 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 ...

**3**

votes

**1**answer

297 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

139 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

94 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

104 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

234 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 ...

**-1**

votes

**0**answers

109 views

### The shape of certain (commutative) topological space

Edit: According to the comment of Simon, I revise the question as follows:
Let $A=C_{0}(\mathbb{R}^{2})$. Let $B$ be the Banach subalgebra of $A$ generated by all $f\in ...

**0**

votes

**0**answers

15 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

75 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

215 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

204 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

51 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

82 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 $A\cap B=\{0\}$?

**1**

vote

**0**answers

48 views

### question about a genralized Skorokhod topology

Let $D:=D([0,1], R)$ be the space of all cadlag functions defined on $[0,1]$. Now we have the known Skorokhod topology defined by: $\forall f, g\in D$
...

**2**

votes

**0**answers

94 views

### Group topologies on $\Bbb Z$ with dense open sets in $\Bbb T$

Let $\Bbb Z$ be embedded in the circle group $\Bbb T$ by an irrational rotation and regard $\Bbb Z$ as a subgroup/subspace of the topological group $\Bbb T$.
Are there group topologies $\mathcal A$ ...

**4**

votes

**2**answers

81 views

### scott continuity, sub additivity

Let $(X, \sqsubseteq_x)$ and $(Y, \sqsubseteq_y)$ be two posets and let $\delta_x:X \to X$ and $\delta_y:Y \to Y$ be two closure operators (monotone, inflationary, idempotent). Then, a monotone ...

**0**

votes

**0**answers

114 views

### extension of function in an abstract metric space

my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set ...

**2**

votes

**1**answer

95 views

### Characterization of a subset of [0,1] $III$

I have a question related to the previous one.
Characterization of a subset of [0,1] $II$
Let $T\subseteq [0,1]$ be some subset closed under lower limit topology, i.e.
$t_n$ is said to converge to ...

**2**

votes

**0**answers

34 views

### Removable sets for simply connectedness of a differentiable manifold

I am sorry that my question might be stupid for experts, but I really do not know the answer.
Let $M$ be a smooth $n$-manifold. We assume that there exists a distance $d$ on $M$ such that $(M,d)$ is ...

**3**

votes

**1**answer

78 views

### cartesian product rigidity for the punctured open disc

Q1: Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to ...

**1**

vote

**1**answer

106 views

### Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$.
So if $x_{1},x_{2}\in A\setminus B$, but they are ...

**1**

vote

**1**answer

151 views

### Relative Compactness vs Way Below in Locally Compact Hausdorff Spaces

Let $Y$ be a subset of a locally compact Hausdorff topological space $X$ and consider the following properties.
$\overline{Y}$ is compact.
Every open cover of $X$ has a finite subcover of $Y$.
...

**4**

votes

**1**answer

69 views

### metrizable neighborhoods of compact subsets

This is a question about general topology:
Assume we are given a first countable Hausdorff space and a compact subset K.
Is it possible to find a countable basis of open neighborhoods of K ?
...

**9**

votes

**1**answer

548 views

### Is every continuous function measurable?

This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow.
In non-Hausdorff topology it is standard to ...

**7**

votes

**1**answer

238 views

### Counting norms on an infinite dimensional vector space

It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...

**1**

vote

**0**answers

80 views

### question about the tightness of probability measures for a general topological space

Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on ...

**5**

votes

**2**answers

333 views

### If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
The answer is negative, and in the interests of self-contained ...

**3**

votes

**0**answers

52 views

### Continuous extension of Riemann maps and the Caratheodory-Torhorst Theorem

If $G\subsetneq\mathbb{C}$ is a simply-connected plane domain, then by the Riemann mapping theorem there is a conformal isomorphism $\newcommand{\D}{\mathbb{D}}\varphi:\D\to G$, where $\D$ is the unit ...

**39**

votes

**3**answers

2k views

### If any open set is a countable union of balls, does it imply separability?

If a metric space is separable, then any open set is a countable union of balls. Is the converse statement true?
UPDATE1. It is a duplicate of the question here
...

**6**

votes

**1**answer

227 views

### Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$

I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow ...

**3**

votes

**0**answers

95 views

### The Klee Trick for subsets of $\mathbb{R}^3$

I asked the question Is dimension given by the Klee trick ever sharp?
That question remains unanswered, so I thought I might ask a slightly more concrete question along those lines.
Given a metric ...

**4**

votes

**2**answers

91 views

### Measure on hyperspace of compact subsets

For a Polish space $X$, let $K(X)$ be the set of compact subsets of $X$. Given the topology with basis $\{K\in K(X):K\subset U_0, K\cap U_1\neq\emptyset,\ldots,K\cap U_n\neq\emptyset\}$ for open sets ...

**4**

votes

**1**answer

158 views

### Totally disconnected subspaces

This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then.
Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it ...

**0**

votes

**1**answer

66 views

### Which Hyperspace Topologies Yield Topological Lattices?

At least on a continuum, the binary operations of intersection and union are Vietoris-continuous. But the Vietoris topology only applies the the collection of NONEMPTY closed subsets, and this means ...

**8**

votes

**0**answers

674 views

### Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$.
It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice [1].
Is $(\mathcal L,\subseteq)$ distributive?
$$~$$
...