Questions tagged [gn.general-topology]

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

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Connected components $0-1$ matrices

Let $M$ be a $0-1$ matrix. Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where ...
Turbo's user avatar
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4 votes
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162 views

Remainders in compactifications of completely metrizable spaces

Given a Tychonoff (topological) space $X$, we know that $X$ admits a Hausdorff compactification $cX$. We say the remainder of $X$ in this compactification is $rX=cX\setminus X$. One natural question, ...
Shehzad Ahmed's user avatar
2 votes
0 answers
169 views

Classify spaces that make extension theorems hold

Recall a Polish space is a completely metrizable separable space. Say a Polish space $Y$ is a terminal space if for any Polish space $X$ and any closed $C \subseteq X$, one can extend a continuous ...
Jason Rute's user avatar
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10 votes
1 answer
414 views

Topological spaces admitting CAT(1) metrics

Suppose that $X$ is a locally contractible completely metrizable topological space. Is it true that $X$ can be metrized as a (complete) CAT(1) metric space? The only result in this direction I know is ...
Moishe Kohan's user avatar
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11 votes
1 answer
351 views

Nonperiodic points of piecewise-linear homeomorphisms

Suppose $K$ is a compact polytope and $T$ is a piecewise-linear homeomorphism from $K$ to itself. Suppose also that $T$ is not of finite order (that is, for no $n \geq 1$ is it the case that $T^n(x)=x$...
James Propp's user avatar
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1 vote
0 answers
177 views

Topological characterisation of loop spaces

Let $\Omega\colon \mathrm{Top}_*\to\mathrm{Top}_*$ be the loop space functor assigning to each pointed topological space $X$ the pointed space consisting of all based continuous maps $S^1\to X$ ...
johndoe's user avatar
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5 votes
1 answer
209 views

Continuity of taking collapse maps

Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
Federico Cantero's user avatar
4 votes
3 answers
583 views

Infinite topological spaces such that every subset is a retract

Let $X$ be an infinite set and let $(X,\tau)$ be a topological space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does this ...
Dominic van der Zypen's user avatar
1 vote
1 answer
264 views

Hausdorff spaces such that every subset is a retract

Let $(X,\tau)$ be a Hausdorff space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does $\tau$ have to be discrete?
Dominic van der Zypen's user avatar
3 votes
1 answer
124 views

Name for this topological property similar to "second countable"

let alpha be a cardinal. I want to look at topological spaces with the property that their topology has a basis with cardinality at most alpha. This property of a topological space certainly has a ...
Tom's user avatar
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Unique representability of bounded distributive lattices

Priestley Duality assigns to every bounded distributive lattice $L$ a compact totally order-disconnected topological space $P(L)$, also called a Priestley space. A poset $(P,\leq)$ is called (...
Dominic van der Zypen's user avatar
3 votes
2 answers
691 views

Topological retraction vs categorical retraction

Let $(X,\tau)$ be a topological space. We say that $A\subseteq X$ is a topological retract if there is a continuous map $r:X\to A$ onto a subspace $A \subseteq X$ such that for all $a\in A$ we have $...
Dominic van der Zypen's user avatar
4 votes
1 answer
277 views

CCC Forcing and $\omega_1$ conditions

I have a question about the proof of the Lemma 7.2 in the paper I. Juhász, P. Koszmider and L. Soukup, A first countable, initially $\omega_{1}$-compact but non-compact space, Topology and its ...
Alberto Levi's user avatar
2 votes
1 answer
327 views

A commutative Banach algebra with an abundance of discountinuous functions

Let $A$ be the algebra of all bounded functions from $[0,\;1]$ to $\mathbb{C}$. For $f\in A,\;$ $\omega_{f}$ is the standard oscillation function.. Each of the following two (equivalent) norms on ...
Ali Taghavi's user avatar
0 votes
1 answer
177 views

$Ax=b$ in a function space

Let $X$ be compact Hausdorff topological space, $C(X)$ denote the algebra of complex-valued continuous functions on $X$, $b\in \mathbb{C}^m$, $\mathbf{A}\in C(X)^{m\times n}$, for all $x\in X$, $b\...
Amol Sasane's user avatar
3 votes
2 answers
552 views

Is the defining bijection for a pullback of topological spaces a homeomorphism?

I work in the category of CGWH spaces enriched over themselves. If a space $P$ is the pullback of $A \rightarrow B \leftarrow C$, then for every space $T$ the canonical map $$Top(T,P) \rightarrow Top ...
Alexander Körschgen's user avatar
13 votes
1 answer
666 views

Strongly rigid Hausdorff spaces

A space $(X,\tau)$ is called rigid if $\textrm{Aut}(X)=\{\textrm{id}_X\}$. We say $(X,\tau)$ is strongly rigid if for every continuous map $f:X\to X$ we have that $f = \textrm{id}_X$ or $f$ is ...
Dominic van der Zypen's user avatar
14 votes
2 answers
1k views

Hausdorff spaces with trivial automorphism group

Is the singleton space the only Hausdorff space $X$ such that the set of automorphisms $\varphi: X\to X$ equals $\{\textrm{id}_X\}$?
Dominic van der Zypen's user avatar
3 votes
0 answers
97 views

Paracompact and countably compactly generated spaces

A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces. Are countably compactly generated spaces paracompact spaces? Do we have ...
Kami sh's user avatar
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1 vote
0 answers
457 views

Separability of the space $C(C[0, 1], \mathbb{R})$

Let $E=C([0, 1])$ be the space of all real-valued continuous functions on $[0, 1]$, equipped with the uniform norm. $C(E)$ stand for the continuous real-valued functions on $E$. I am wondering that ...
gregarki khayal's user avatar
9 votes
0 answers
599 views

Two questions about universally measurable sets

I have two questions about universally measurable sets: (1) Is there a universally measurable set of reals which does not have the Baire property? (2) Is there a universally measurable set of reals ...
Ashutosh's user avatar
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3 votes
0 answers
399 views

Generalization of Jordan Curve Theorem

Jordan Curve Theorem says that any plane continuum homeomorphic to $\mathbb{S}^1$ separates the plane into exactly two components. Now "Let $\alpha$ and $\beta$ be two homeomorphic plane continua. ...
Hesam's user avatar
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-3 votes
3 answers
729 views

Riemann Mapping Theorem in Higher Dimensions for Continuous funcions [closed]

Is there any analogue for Riemann Mapping Theorem(!) in higher dimensions? Or a much simpler question, is it true that every open subset of $\mathbb{R}^3$ with zero homology in dimensions 1 and 2 is ...
Hesam's user avatar
  • 615
23 votes
1 answer
696 views

Which ordered fields are homeomorphic to their power?

It is well known that $\mathbb{R}^2\ncong \mathbb{R}$. It is also known that $\mathbb{Q}^2\cong \mathbb{Q}$. It is a corollary to Sierpiński's theorem which states that every countable metric space ...
Asaf Shachar's user avatar
  • 6,611
2 votes
0 answers
104 views

Selecting dense diagonals in $\Bbb T^2$

Let $p$ be a prime number and let $G=\bigcup_{n\in \Bbb N}\{\exp(k\frac{2\pi i}{p^n})\mid k\in \Bbb Z\}$ be a Prüfer group. For homomorphisms $f,g:G\to G$ let $H_{f,g}=\{(f(x),g(x))\mid x\in G\}$. ...
Minimus Heximus's user avatar
4 votes
0 answers
146 views

A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor. Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence of ...
Ali Taghavi's user avatar
2 votes
1 answer
324 views

How many pairwise non-homeomorphic compact, zero-dimensional topologies are there on $\mathbb{N}$?

To make the question more precise: We call a topological space $(X,\tau)$ zero-dimensional if for $x\neq y \in X$ there is a clopen set $U\subseteq X$ with $x\in U, y\notin U$. Let $\mathcal{C}$ be ...
Dominic van der Zypen's user avatar
5 votes
1 answer
237 views

Maps between spaces of non-empty compact subsets with the Hausdorff distance (reference request)

Let $X, Y$ be metric spaces, and let $PX$ (resp. $PY$) be the set of all non-empty compact subsets of $X$ (resp. $Y$) with the Hausdorff metric. A continuous map $f\colon X\to Y$ induces a continuous ...
Federico Cantero's user avatar
5 votes
2 answers
189 views

another question about connected open sets in $R^2$

Before posting this question,I just asked a similar question:a question about connected open sets in $R^2$. I got several nice answers.Now I want to ask: Let $U$ be a nonempty connected open set in $...
user173856's user avatar
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10 votes
2 answers
352 views

a question about connected open sets in $R^2$

Let $U,V$ be two nonempty connected open sets in $\mathbb{R}^2$ and $U\subsetneqq V$.I want to ask if there must exist an open ball $B\subset V$ such that $B\not\subset U$ and $B\cap U$ is a nonempty ...
user173856's user avatar
  • 1,987
4 votes
1 answer
308 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 $X$?...
Ali Taghavi's user avatar
4 votes
3 answers
1k 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 (...
Adam's user avatar
  • 445
6 votes
1 answer
282 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 ...
A Rock and a Hard Place's user avatar
7 votes
1 answer
234 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 ...
user avatar
2 votes
1 answer
460 views

totally disconnected sets and homeomorphisms [closed]

For every totally disconnected perfect subset S in the plane one finds a homeomorphism of the plane onto itself mapping S onto the ternary Cantor set. This is an exercise in a book by Engelking and ...
ray's user avatar
  • 687
12 votes
1 answer
630 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 ...
Dominic van der Zypen's user avatar
2 votes
0 answers
331 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, ...
CodeGolf's user avatar
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2 votes
1 answer
85 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 $g : D^n \to \mathbb{R}^n$ ...
Sridhar Ramesh's user avatar
3 votes
1 answer
148 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. ...
Al Nejati's user avatar
  • 133
2 votes
1 answer
767 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 ...
CodeGolf's user avatar
  • 1,837
2 votes
1 answer
134 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 ...
CodeGolf's user avatar
  • 1,837
1 vote
0 answers
135 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 ...
Ali Taghavi's user avatar
4 votes
1 answer
118 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 ...
Geno Whirl's user avatar
6 votes
2 answers
498 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 ...
user avatar
6 votes
0 answers
104 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 ...
Taras Banakh's user avatar
  • 40.8k
0 votes
1 answer
278 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 ...
user61430's user avatar
3 votes
0 answers
113 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 ...
Niek de Kleijn's user avatar
5 votes
1 answer
468 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 ...
Tom's user avatar
  • 967
3 votes
0 answers
311 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 ...
Dominic van der Zypen's user avatar
-1 votes
1 answer
78 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 $K$? ...
someone's user avatar

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