Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
3,421
questions
0
votes
0answers
23 views
Relationship between tangent cone and support of density operators
Let $H$ be a separable Hilbert space. Let $(T,\Vert \cdot \Vert_1)$ be the Banach space of self-adjoint trace-class operators on $H$ with the trace norm $\Vert \cdot \Vert_1$. Finally, let $D$ be the ...
-1
votes
1answer
59 views
Proving neighborhood of a compact product space contains a sub-neighborhood formed by taking product [closed]
I am self studying basic topology and have trouble proving the following question.
If $A$ and $B$ are compact, and if $W$ is a neighborhood of $A \times B$ in $X \times Y$, find a neighborhood $U$ of ...
3
votes
1answer
304 views
What is Bouziani space and what are its applications in mathematics?
I have accrossed a new topological space seems were derived from Hilbert Space and it used to solve some boundary value problem for PDE and ODE , Inspired by this paper (page 4, Definition 3.1) , The ...
8
votes
0answers
133 views
Is every Baire metric space a complete metric space in disguise?
I am currently giving lectures in real analysis and a student asked an interesting question I couldn't answer, so I'm posting it here:
Let's say that a metric space $X$ is Baire if every countable ...
1
vote
0answers
51 views
A weakly sequentially continuous operator which is not weakly continuous
I'm reading some papers where the condition of weak sequential continuity is crucial instead of the weak continuity.
So, let
$T$ an operator between a Banach space $X$ and itself.
$T$ is weakly ...
2
votes
0answers
98 views
Products of cones and cones of joins
The join of $A$ and $B$ is the pushout of the diagram
$$
CA \times B \gets A\times B \to A\times CB,
$$
which can be formulated in either the pointed or unpointed topological
category. This pushout is ...
-1
votes
0answers
87 views
On Čech-complete space
I'm reading an article of topology and i came across a Properties :
Properties :
Closed subspaces and arbitrary products of Čech-complete spaces are Čech-complete
Every Čech-complete space is a Baire ...
7
votes
1answer
214 views
The space of skew-symmetric orthogonal matrices
Let $M_n \subseteq SO(2n)$ be the set of real $2n \times 2n$ matrices $J$ satisfying $J + J^{T} = 0$ and $J J^T = I$. Equivalently, these are the linear transformations such that, for all $x \in \...
4
votes
1answer
82 views
Covering of discrete probability measures
Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
4
votes
1answer
63 views
Continuous selection parameterizing discrete measures
Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
11
votes
0answers
297 views
Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
9
votes
0answers
169 views
Homotopical characterization of manifolds
Let $X$ be a compact metrizable topological space of covering dimension $4$.
Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...
1
vote
0answers
34 views
A local base for space of probability measures with Prohorov metric
Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
1
vote
1answer
47 views
irreducible compact set vs prime compact set
Let $X$ be a topological space.
A compact set $K$ is called irreducible if for any two compact subsets $K_1$, $K_2$ of $K$ with $K$ is equal to the union of $K_1$ and $K_2$, then $K$ is equal to $K_1$ ...
2
votes
0answers
61 views
Topology of the set of polynomials with bounded real algebraic varieties (inside the v. s. of polynomials in $n$ variables and up to degree $d$)
Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector ...
-2
votes
0answers
139 views
Fundamental group and countability
A friend of mine on a Discord server talked about an exercise she had to do (she's in master's degree): prove that you can put an uncountable number of disjoint "5" in the euclidean plane $\...
59
votes
4answers
4k views
What are good mathematical models for spider webs?
Sometimes I see spider webs in very complex surroundings, like in the middle of twigs in a tree or in a bush. I keep thinking “if you understand the spider web, you understand the space around it”. ...
17
votes
1answer
309 views
Axiom of Countable Choice and meager sets
Let us recall that the Axiom of Countable Choice (denoted by ACC) says that the countable product $\prod_{n\in\omega}X_n$ of nonempty sets $X_n$ is nonempty.
It is easy to see that ACC implies that ...
2
votes
0answers
73 views
Is the space of metric topologies over a given set dense (in the order sense)?
Suppose that $S$ is an infinite set and that $\alpha$ and $\beta$ are metrics over $S$ such that the topology induced by $\alpha$ is everywhere strictly finer than the metric induced by $\beta$, ...
1
vote
0answers
91 views
Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
0
votes
0answers
137 views
Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
7
votes
0answers
147 views
Does there exist a complete metric space which is Rothberger (or Menger) but not Hurewicz?
A topological space $X$ is said to be a
Menger space if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there is a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a ...
-1
votes
0answers
60 views
A finite union of convex bounded open subsets of $S^{n}$ has finitely generated cohomology groups [migrated]
Let $V_{1},...,V_{n}\subset S^{d}$ be convex, bounded, open subsets of $S^{d}$. Show that $H^{d-1}(\cup_{i=1}^{n} V_{i})$ is finitely generated.
According to a solution here Complement of a finite ...
4
votes
0answers
151 views
Sierpinski's characterization of $F_{\sigma\delta}$ spaces
According to [2]: Let $X$ be a space. We call a system $(X_s)_{s\in T}$ a Sierpinski
stratification of $X$ if $T$ is a nonempty tree over a countable alphabet and $X_s$ is a closed subset of $X$ for ...
1
vote
1answer
123 views
Function series of normal lower semi-continuous functions
For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is
defined as follows:
$
f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in
U\right\} :U\in ...
0
votes
0answers
47 views
Are these boolean subalgebras always the clopen sets of some topology?
What would one call a family $\mathcal{F}$ of subsets $X$ such that:
$$(1):X\in\mathcal{F}$$
$$(2):A,B\in\mathcal{F}\implies A\setminus B\in\mathcal{F}$$
$$(3):P\subseteq\mathcal{F}\text{ is a ...
33
votes
4answers
3k views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
6
votes
2answers
241 views
Non-sequential spaces in the wild
TLDR: What are examples of (function-)spaces that are not sequential? When does this matter?
As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
1
vote
1answer
125 views
Regularity and ultrafilter
I read the following result in an article.
Let $X$ be a regular space. Let $\mathcal{M}$ be free closed
ultrafilter on $X$. Set $\mathcal{U=}\left\{ U:U\text{ is open and there
exists a }F\in \mathcal{...
0
votes
1answer
139 views
P-filter property?
Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega_i$ where $\omega_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup_i ...
6
votes
2answers
153 views
Nonhomeomophic spaces with homeomorphic mapping cones
It is natural to ask if it is possible for the mapping cone $X\cup_\alpha CA$
to be homeomorphic to the mapping cone $X\cup_\beta CB$ with $A$ and $B$
nonhomeomorphic. Is there a standard go-to ...
1
vote
1answer
139 views
Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
2
votes
1answer
126 views
Set of null-sequences is not $\sigma$-compact
I am interested in a reference for the following fact (or a similar result).
PROPOSITION. Let $X$ denote the set of real null sequences; i.e., the set of $(a_n)_{n=0}^{\infty}$ with $a_n\to 0$, with ...
4
votes
1answer
136 views
Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?
Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
0
votes
0answers
93 views
Problem of Thickening an Arc in a Topological $ 2 $-Manifold
Let $ M $ be a topological $ 2 $-manifold (possibly with boundary), $ C $ an arc in the interior of $ M $ (i.e., an injective continuous function from $ [- 1,1] $ into $ \operatorname{Int}(M) $), and $...
2
votes
0answers
70 views
Compact metrizable contractible locally contractible topological space of finite covering dimension is a CW complex
Let $X$ be a compact metrizable contractible locally contractible topological space of finite covering dimension. Is $X$ homeomorphic to a CW complex?
1
vote
1answer
50 views
Projecting Graph of a Function acted on by a homeomorphism
Let $X,Y$ be compact, connected, simply-connected, and separable, metric spaces each with at-least $2$-points, and let $f,g:X\rightarrow Y$ be continuous functions. Does there always exist a ...
6
votes
1answer
308 views
Any continuous map is homotopic to one assuming fixed values at finitely many points
Let $X$ and $Y$ be topological spaces. Assume $X$ is locally contractible and has no dense finite subset. Assume $Y$ is path-connected.
Given $n$ pairs of points $(x_i, y_i)$ where $x_i\in X$ and $y_i\...
7
votes
0answers
122 views
adding one point from the Stone-Cech compactification
Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
2
votes
0answers
56 views
Separating a certain planar region with an open set
I have a fairly specific question related to plane separation properties. I couldn't quite see how to use Phragmen–Brouwer properties to answer it because those kind of results generally apply to ...
4
votes
0answers
159 views
When every closed and connected subset is path connected
Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?
-1
votes
2answers
169 views
Intrinsically defining smooth/continuous/analytic functions
In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
2
votes
1answer
114 views
Are there minimal topological conditions on a space 𝑋 for it to have a countable separating set?
Are there minimal topological conditions on a space $X$ for it to have a countable separating set?
A separating set here is a set $D \subset C(X)$ (where $C(X)$ is the space of continuous functions ...
4
votes
0answers
102 views
'Monodromy' for relative homology group
Let $A$ and $X$ be topological manifolds. Denote by $\mathbb {Emb}(A,X)$ the space of all topological embeddings $A\to X$.
A loop $f_s:A\to X$ ($s\in[0,1]$) in $\mathbb{Emb}(A,X)$ should give rise to ...
2
votes
1answer
56 views
Equicontinuity-like property of a convex compact set
Let $X$ be a Tychonoff topological space and let $x\in X$. Let $B\subset C(X)$ be convex and compact in the topology of pointwise convergence, and such that $f(x)=1$, for every $f\in B$.
Is there an ...
5
votes
2answers
124 views
Local cross-sections for free actions of finite groups
Let $G$ be a finite group, let $X$ be a locally compact Hausdorff space, and let $G$ act freely on $X$. It is well-known that the canonical quotient map $\pi\colon X\to X/G$ onto the orbit space $X/G$ ...
4
votes
1answer
260 views
Mapping $\mathbb P$ onto $\mathbb Q ^\omega$
Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-...
4
votes
1answer
230 views
Functoriality of Atiyah-Hirzebruch spectral sequence - Reference Request
I'm interested in a text book reference on the functoriality of the Atiyah–Hirzebruch spectral sequence. The only reference I found are these lecture notes by Kupers (link should lead to the target ...
3
votes
0answers
42 views
Critical Growth of Dimension for Dense Cover by Linear Subspaces
Let $X$ be a separable Banach space of dimensional $>2$. When does there exist a sequence positive integers $\{N_n\}_{n \in \mathbb{n}}$ such that
For any sequence of distinct finite-dimensional ...
5
votes
1answer
147 views
Arc connectedness of product spaces
Is arc connected-ness well-behaved with respect to products?
That is -
$\prod X_\alpha$ is arc connected iff $X_\alpha$ is arc connected $\forall \alpha$
In this question on MathStackexchange, an ...
