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

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0
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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
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1answer
101 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
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1answer
101 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
1answer
111 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 ...
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0answers
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
2answers
217 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
1answer
147 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
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1answer
183 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
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0answers
166 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
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1answer
488 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
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1answer
250 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 ...
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2answers
164 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
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0answers
178 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
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0answers
80 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
1answer
135 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
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0answers
56 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
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0answers
161 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
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2answers
231 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 : ...
2
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1answer
164 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
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0answers
68 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
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0answers
61 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
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0answers
66 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
1answer
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
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2answers
215 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
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1answer
73 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
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0answers
319 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
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1answer
228 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
2answers
337 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
1answer
494 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
2answers
187 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
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1answer
171 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
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2answers
388 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?
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1answer
124 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
1answer
149 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
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3answers
386 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
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0answers
83 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
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0answers
108 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
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0answers
181 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
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1answer
199 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 ...
13
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2answers
345 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
1answer
307 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
1answer
197 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
1answer
105 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
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0answers
159 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
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2answers
484 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?
14
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2answers
959 views

Generalization of Darboux's Theorem

Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$. Does any of the following generalizations Let ...
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0answers
75 views

Topologies on spaces of linear sections

Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable. Let $f : X ...
12
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7answers
683 views

Examples of toposes for analysts

I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand. Can you provide some examples ...
2
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1answer
152 views

characterization of the unit disk

We know of a charcterization of spaces homeomorphic to [0,1], as being metric continua with 2 noncut points. We have as well a characterization of spaces homeomorphic to the unit circle. I can't find ...
6
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1answer
129 views

Constructing a odd homeomorphism between $A$ and $S^n$

I have asked this question here seven months ago and until now I got no answer. Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is ...