**2**

votes

**1**answer

110 views

### A souped-up version of a question asked previously about uncountable subsets of topological spaces

Let T be an uncountable Hausdorff space. The following property of T will be referred to as "property P". If S is any uncountable subset of T, then the set of all points of S that are not limit points ...

**16**

votes

**2**answers

457 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

**6**

votes

**3**answers

393 views

### Do any Stone-like dualities have some self-dualities hidden inside them?

This question originated from the observation that in most cases when one has duality of structured sets induced by a dualizing set-with-two-structures $D$, both sides of the duality are substructures ...

**3**

votes

**1**answer

193 views

### Strange (?) definition of the spectrum

Suppose that $A$ is a commutative, unital $C^*$-algebra. Then it is isomorphic to $C(X)$ for some compact Hausdorff topological space $X$. $X$ can be identified as the space of all unital ...

**0**

votes

**0**answers

30 views

### Is the multiplicity of a Sobolev mapping alwasys locally essentially bounded？

I have the following question:
Let $f:\Omega\to \mathbb{R}^n$ be a (sense-preserving) continuous Sobolev mapping in $W^{1,n}$, where $\Omega$ is a domain in $\mathbb{R}^n$. By Sard's theorem for ...

**0**

votes

**1**answer

89 views

### Homotopy with non piece-wise linear boundary

in the middle of a long proof I encounter the following problem.
Let $E$ be a closed and convex set in $\mathbb R^n$ such that for all $\vec x\in E$ it holds that $\sum_ix_i=1$. (We can understand ...

**0**

votes

**1**answer

93 views

### Rational points in the Alexandroff line

Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line.
Is $X$ a metrizable space?

**2**

votes

**1**answer

108 views

### Connectedness properties of groups of homeomorphisms

Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological ...

**0**

votes

**0**answers

40 views

### Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...

**3**

votes

**2**answers

203 views

### Non-Polish Lebesgue probability space?

Any Lebesgue probability space is mod. 0 isomorphic to some Polish probability space (with $\sigma$-algebra the completion of Borel algebra, and some Borel probability). I would like to see an example ...

**6**

votes

**1**answer

131 views

### A question about cardinal functions

In the context of topological groups, one has $w(X)=d(X)\chi(X)$, where $X$ is a topological group and $w$, $d$ and $\chi$ are the weight, the density and the character, respectively. Since $d(X)\leq ...

**1**

vote

**1**answer

176 views

### Tietze's extension theorem for compact subspaces

The topological question:
Are there Hausdorff topological spaces $X$ which are compactly generated (=Kelly spaces = $k$-spaces, that is, a subset is closed if its intersection with every compact set ...

**6**

votes

**0**answers

143 views

### Zariski-homeomorphisms

This question is motivated by two questions at MO and
at MSE.
I am interested in homeomorphism types of (irreducible) complex-projective varieties with respect to the Zariski topology. Any two ...

**15**

votes

**1**answer

468 views

### Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...

**0**

votes

**1**answer

241 views

### Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...

**-3**

votes

**2**answers

139 views

### The boundary of this set is piecewise smooth? [closed]

Consider a sequence of open sets in $R^n$: $\Omega_1 \supset \Omega_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean ...

**2**

votes

**0**answers

148 views

### Approximation of continuous functions by Lipschitz functions in the topology of uniform convergence on compact sets

I was involved into this subject when I answered
this
question from MSE. Trying to generalize my answer, I am thinking about a following
Question. Let $X$ and $Y$ be metric spaces. When each ...

**2**

votes

**0**answers

79 views

### The Haar integral on uniform spaces

Does anybody know what the current research status is on the topic of Haar measures on uniform spaces? I'm specifically interested in compact uniform spaces and its corresponding Haar probability.
As ...

**0**

votes

**1**answer

134 views

### US does not imply AB

We say that a topological space $X$ is:
$AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A ...

**2**

votes

**0**answers

48 views

### A construction with Hyperspace of continums

Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...

**2**

votes

**0**answers

158 views

### Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere.
Unfortunately I do not have his book but I want to know is this theorem true without ...

**5**

votes

**2**answers

226 views

### Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...

**0**

votes

**0**answers

193 views

### Problem on infinite dimensional metric space, with rigidity assumption

By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption.
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying :
...

**2**

votes

**1**answer

126 views

### A question about small cardinals related to Michael's Problem

I'm studying some applications of small cardinals related to the Michael's Problem. Recall that we say that a space $X$ is a Michael space if X is a regular Lindelöf space such that $X\times ...

**0**

votes

**0**answers

67 views

### What do sparse sets in a norm topology look like in the weak* topology?

I'm wondering if a very "sparse" set in a normed vector space can look connected in the weak* topology. Specifically,
Let V be a Banach space, V* its dual, and X a (uncountable) subset of the unit ...

**0**

votes

**0**answers

55 views

### hyperspaces and selection principals

Two things bother me for which I haven't found an answer yet:
1.Is anyone familiar with an example of a topological space $X$, in which the hyperspace $2^X$ with the upper Fell topology is ...

**2**

votes

**0**answers

56 views

### A categorical analogue of Debreu's independent factors theorem

Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...

**3**

votes

**1**answer

206 views

### Density of linear functionals in $L^2$

Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals ...

**1**

vote

**2**answers

212 views

### Existence of non-locally constant functions

Given a nondiscrete compact Hausdorff space $K$, does there always exist a real-valued function $f$ on $K$ that is not locally constant? Why/why not?
In http://arxiv.org/abs/math/9505204 the authors ...

**0**

votes

**0**answers

49 views

### subspace in pseudotopological space

Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...

**2**

votes

**0**answers

219 views

### Convex hulls of compact sets

Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?

**4**

votes

**1**answer

226 views

### What is the Stone–Čech compactification of a dense set of $\beta N \setminus N$?

Is the Stone–Čech compactification of a dense $G_\delta$-set $X \subset\beta N \setminus N$ homeomorphic to $\beta N \setminus N$? Here, $\beta N \setminus N$ is the complement of $N$ in the ...

**3**

votes

**2**answers

336 views

### The quotient of $\mathbb{R}^{n}$ by a closed subset

Let $A$ be a closed subset of $\mathbb{R}^{n}$. Can the quotient space $\mathbb{R}^{n}/A$ be embedded in some Euclidean space $\mathbb R^{m}$? In particular, assume that $A$ is an algebraic variety of ...

**17**

votes

**1**answer

482 views

### Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?

It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...

**6**

votes

**2**answers

179 views

### Fixed points and their continuity (2)

Yesterday I asked a question about fixed point. Here is the link.
In summary, the question was,
Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic ...

**6**

votes

**1**answer

165 views

### Fixed points and their continuity

Let $f : I^2 \to I$ be a continuous map, where $I := [0,1]$ is the unit interval. It is a basic fact that for each $y\in I$, the function $x \mapsto f(x,y)$ admits a fixed point. I want to ask whether ...

**11**

votes

**2**answers

382 views

### subsets of groups which have to be closed no matter what

One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?

**-1**

votes

**1**answer

122 views

### Is the countably infinite product of locally convex topological vector spaces locally convex?

Let $(X,\tau)$ be a locally convex topological vector space and denote the product space
$$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$
If we endow $X^{\infty}$ ...

**3**

votes

**1**answer

148 views

### Constructible subset of constructible set

Let $X$ be a topological space. Let $F \subset E \subset X$ be subsets. Assume that $E$ is constructible in $X$ and that $F$ is constructible in $E$. Is it true that $F$ is constructible in $X$?
We ...

**9**

votes

**3**answers

369 views

### Is every T0 2nd countable space the quotient of a separable metric space?

Suppose the space $X$ has a countable basis and $X$ is $T_{0}$. Must there exist a separable metrizable space $Y$ and a quotient map q:$Y \rightarrow X$?
(Some surrounding facts:
Every metrizable ...

**1**

vote

**0**answers

75 views

### equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)

Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$,
$$
...

**2**

votes

**0**answers

107 views

### Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups
by Marshall Hall Jr. in which profinite groups are defined for the first time.
There he defines on p. 129:
...

**3**

votes

**0**answers

177 views

### Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...

**1**

vote

**1**answer

187 views

### Interior of a dual cone

Let $K$ be a closed convex cone in $\mathbb{R}^n$. Its dual cone (which is also closed and convex) is defined by $K' = \{ \phi\ |\ \phi(x) \geq 0,\ \ \forall x \in K\}$.
I know that the interior of ...

**12**

votes

**2**answers

317 views

### compact-open topology on $B(H)$

In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...

**3**

votes

**1**answer

304 views

### Is there a simple topological proof for a topological theorem about $S^2$?

Consider the problem of coloring each point of $S^2$ with one of two colors (say "black" or "white") so that among any three points of $S^2$ which are the vertices of an equilateral spherical triangle ...

**9**

votes

**1**answer

195 views

### Is it always possible to “encircle” exactly $n$ points in an infinite subset of $\mathbb{R}^d$ without limit points?

Let $d$ be a positive integer, and let $\mathbb{R}^d$ be endowed with the
Euclidean metric. Given an infinite set $S \subset \mathbb{R}^d$ without limit points
and a positive integer $n$, is there ...

**2**

votes

**1**answer

104 views

### Projective limit and connected components

Let $E$ be a topological space. Let $\mathcal{K}$ be the set of the compact subsets of $E$.
$(E-K)_{K \in \mathcal{K}}$ is a projective system, because if $K,K'$ are two compacts, there are two ...

**9**

votes

**0**answers

157 views

### Multiplicity of ball covering

Background. My questions are motivated by the following:
A. Conway and Sloane in "On the covering multiplicity of lattices" (Discrete
and Computational Geometry, 8 (1992) 109-130) considered the ...

**8**

votes

**2**answers

451 views

### Totally disconnected locally compact Hausdorff spaces

Can any totally disconnected locally compact Hausdorff space be written as a disjoint union of subsets that are both compact and open?
If this is true, does anyone know of a good reference?