Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,602 questions
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Equivariant zero dimensional extension recovering a given measure
Let $X$ be a compact metrizable space and $\alpha: \mathbb{Z}^d\curvearrowright X$ a continuous group action. Then it is well known that there exists a zero dimensional compact space $Y$, an action $\...
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Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?
Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2?
Thanks
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Is there a compact group of countably infinite cardinality?
Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) ...
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Can connectedness of fibers of a smooth map be checked on a dense set?
Suppose $f: M\to N$ is a smooth map between two smooth manifolds, with $M$ compact and connected, and suppose there is a dense subset of $f(M)$ where each fiber is connected, then each fiber of $f$ is ...
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Non-compact structure group and compactly supported gauge transformations
Let $\pi\colon P\to X$ be a locally trivial principal $G$-bundle over a Hausdorff paracompact space $X$, where $G$ is a topological group (we work in the category of topological spaces, as I do not ...
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Spaces that never separate the Hilbert cube
I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this ...
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Isomorphisms between topological vector spaces [closed]
Let $f : X \to Y$ be a continuous map of complete topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A \...
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Is it impossible for the dimension of a topological space to increase under a smooth map?
First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...
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Is there a monad on Set whose algebras are Tychonoff spaces?
Compact Hausdorff spaces are algebras of the ultrafilter monad on Set.
Is the category of Tychonoff spaces also monadic over Set?
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Filters and intersection of two binary relations
Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} =...
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Question about product topology
Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$.
Is $S\times S$ homeomorphic to $S$?
By Luzin ...
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Can distinct open knots correspond to the same closed knot?
A topological ("closed") knot is an embedding of a circle in $\mathbb{R}^3$. It's possible for a knot to be distinct from the unknot because there are no free ends to move around and untie the knot. ...
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Agreement of two topologies on a linear space
I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
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How much "Morse theory" can be accomplished given only a continuous transformation of a space?
If $M$ is a Riemannian manifold and $f:M\to \mathbb{R}$ a Morse-Smale function (which is just a rigorous way to say "generic smooth function"), then Morse theory essentially recovers the manifold ...
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Topological Generalization of Whitney's Extension Theorem
From Wikipedia:
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if $A$ is ...
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Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
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Orbit spaces of involutions on spheres
I'm studying the following problem: Let
$({\mathbb S}^N,\theta)$ be the $n$-sphere (in ${\mathbb R}^{N+1}$) endowed with the antipodal action $\theta:(x_0,\ldots,x_N)\to (-x_0,\ldots,-x_N)$;
$({\...
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When factors may be cancelled in homeomorphic products?
It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^{...
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Homeomorphisms and disjoint unions
Let $X$ and $Y$ be compact subsets of $\mathbb{R}^n$. Assume that $X \sqcup X \cong Y \sqcup Y$ (here $X \sqcup X$ is the disjoint union of two copies of $X$, considered as a topological space, and ...
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3 manifolds with diffeomorphic unit tangent bundles
What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
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How to transform a plane into a sphere? [SOLVED] [closed]
Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...
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ball in universal cover belongs to the union of actions on a section?
M is an n-dim manifold. $\pi :\tilde M \to M$ the universal cover of M. $\tilde p \in \tilde M$ a lift of p. We choose a measurable section $j:{B_1}\left( p \right) \to {B_1}\left( {\tilde p} \right)$,...
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Is there a contractible bounded homogeneous space?
Let us phrase the question in the title in more detail: I wonder if there exists a metric space $X$ which has at least two points, has finite diameter (in the sense that there is an upper bound for ...
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Topological Characterisation of the real line.
What is a purely topological characterisation of the real line( standard topology)?
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Uniformities generated by metrics.
Any uniformity on a set $X$ is generated by a family of pseudometrics on $X$. So if $(X,\mathcal D)$ is a uniform space there's a set $P$ of pseudometrics on $X$ with
$$\mathcal D=\left< \bigcup_{...
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When is the support of a Radon measure separable?
Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the ...
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How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$?
I can see that results in both ...
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Topology of Asymmetric Symmetric Products
Let $X_1,...,X_m$ be connected, simply-connected CW sub-complexes of a CW complex $X$. Let the symmetric group on $m$ letters, $S_m$, act on $P:=X_1\times\cdots\times X_m$ in $X^m$ by permuting ...
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G-equivariant Whitehead's Theorem
Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume ...
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Analogues of the Weierstrass p function for higher genus compact Riemann surfaces
There was a previous post on the correspondence between Riemann surfaces and algebraic geometry. I want to ask a related but more detailed question.
BACKGROUND:
Engelbrekt gave an overview of how ...
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Does there exist a space X whose suspension is homotopy equivalent to [0,1] rel ends but where X is not contractible?
As pointed out by David White in
when mapping cone is contractible
there exists an acyclic CW-complex $X$ which is not contractible but whose suspension is contractible. Namely, let $a$ and $b$ be ...
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How far is Lindelöf from compactness?
A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
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the example of ccc but not separable
I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.
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The pth power of a distance function is twice continuously differentiable, for $p>2$?
Suppose $\mathcal{O}$ is an open convex connected strict subset in $\mathbb{R}^n$ and define $\beta(x)=dist(x, \mathcal{O})$, for each $x\in\mathbb{R}^n$.
Is $\beta^p$, $p>2$ a twice continuously ...
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Unit sphere in R^\infty is contractible?
Hello,
We know that in Hilbert space it is, but what about these topology:
Let $\mathcal{T}_{\infty}= \{ U \subset \mathbb{R}^{\infty}: \ U \cap \mathbb{R}^n \in \mathcal{T}_n, \ for \ n=1,2,... \} $ ...
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The integers as a sequential but non-first countable topological group
Completely unaware of the Bohr topology, I recently asked whether or not there was a Hausdorff group topology on the integers $\mathbb{Z}$ which made the group fail to be first countable. For me, this ...
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An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request
There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's ...
CommunityBot
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Questions about knot (link) of surface in four dimension
Consider three 2-torus ($S^1*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked ...
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A conjecture on closed discrete subset
I am struggling with this old problem, which is also posted here:
Let $X$ satisfy countable chain condition(abbreviated as CCC) and $X$ has a regular $G_\delta$-diagonal. Then the cardinality of $X$...
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A question on continuous mappings
The question is also posted here.
Let $M=\mathbb{R}$ and $\tau_M=\lbrace U\cup A: U$ open in $\mathbb{R}, A\subset \mathbb{R} \setminus B\rbrace$, where $B$ is a Bernstein set. Then $(M,\tau_M)$ is a ...
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How to visulize surface link in four dimension?
I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...
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A question on star $\sigma$-compact spaces
The question is also posted here.
A topological space $X$ is said to be star $\sigma$-compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a $\sigma$-compact subspace $K$ of $X$ such ...
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A question about indecomposable continua.
The term "continuum" is often used to mean a compact and connected metric space. But it is
also used in a broader sense to mean any infinite, complete, separable and connected metric
space-which is ...
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A question on metrizable space
Q1, Does a metrizable space $X$ with $e(X)=\omega$ (i.e., it has countable extent) which is not lindelof exist?
Q2, Let $X$ be the one point lindefication of a discret space of cardinality $\omega_1$...
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Which topological spaces are (topological) groups?
General literature does not seem to offer a characterisation of topological groups among all topological spaces. Of course, being completely regular (uniform) is necessary, but separation properties, ...
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Topological dimension of quotient group determined by the inverse limit of discrete free monoids
Must the natural quotient group of the inverse limit of a sequence of nested discrete free monoids have topological dimension zero?
The question might well be open, but I would be grateful for news ...
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Topological question about right-lifting property and the evaluation map
Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ ...
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A question on cofinite topology.
Let $X$ be a countably infinite (or larger) set with the cofinite topology. for every $x\in X$ is there exists a family $\xi\subset\tau$ such that $\lbrace x\rbrace=\bigcap\xi
$ ? If the answer is yes,...
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Extending uniformly continuous functions on subspaces to non-metrizable compactifications
I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$.
Furthermore, I do have a uniformly continuous function $f$ on $X$. So ...
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Regular Borel Measures equivalent definition
Please help me understand how the below definition is equivalent to the standard definition of regularity which says that a measure is regular if for which every measurable set can be approximated ...
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