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

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4
votes
1answer
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
0answers
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
0answers
105 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
0answers
58 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
2answers
131 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
1answer
146 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
0answers
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
1answer
180 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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
59 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
1answer
100 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
1answer
72 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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
198 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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
68 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 $r$ ...
7
votes
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
2answers
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
0answers
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
1answer
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 $A\cap B=\{0\}$?