Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,431
questions
3
votes
1
answer
151
views
Embedding of half open half closed $n$-set in $n$-space
Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...
4
votes
2
answers
388
views
Is a local diffeomorphism with nice boundary values a diffeomorphism?
Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.
The only situation I can image ...
0
votes
0
answers
17
views
Is the impression of an ideal boundary point (=end) the union of the impressions of the prime ends of the circle of prime ends associated to this end?
Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (...
0
votes
0
answers
38
views
Generic non-existence of 1. Integral of continuous DS
Let $M$ be a compact manfiold and $F \in \mathcal{X}(M)$. We define a DS on $M$ by
$$\dot{\mathbf{x}}=F(\mathbf{x}(t))$$
In 1 it was shown by Hurley, that a generic diffeomorphism on $M$ does not have ...
1
vote
1
answer
79
views
Reference for k-Hausdorff (in terms of compact T2 images)
In Rezk - Compactly generated spaces a k-Hausdorff property is defined, between weakly Hausdorff and unique sequential limits.
On the other hand, a stronger notion of k-Hausdorff between $T_2$ and ...
0
votes
0
answers
296
views
Proof that a first integral is not a constant function
Let $U$ be an (open) set in $\mathbb{R}^n$. And we are given a set of $m$ basis functions
$$B=\{\psi_i(x): U \rightarrow \mathbb{R}\mid i=1,\ldots,m \}$$
such that all of them are differentiable and ...
2
votes
1
answer
139
views
Variation of concept of a Lusin space
Citing from Wikipedia,
A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space.
Is there a (previously studied) analogous concept of a Hausdorff (...
14
votes
0
answers
415
views
Which functions have all the common $\forall\exists$-properties of continuous functions?
This is an attempt at partial progress towards this question. Meanwhile, Sam Sanders pointed out that my original term was already in use, as were a couple other back-up terms, so ... oh well.
For a ...
6
votes
1
answer
441
views
Which maps of topological spaces have the right lifting property with respect to all split monomorphisms?
Let $p : X \to Y$ be a continuous map. We say that $p$ has the right lifting property with respect to split monomorphisms if, for every space $B$, and every retract $A \subseteq B$, and for every ...
0
votes
1
answer
484
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
21
votes
2
answers
966
views
Can a continuous real-valued function on a large product space depend on uncountably many coordinates?
Is there a reasonably well-behaved topological space $X$ (ideally Polish), a set $\kappa$, and a continuous function $g: X^\kappa\to\mathbb{R}$ that depends on uncountable many coordinates?
If $X$ is ...
8
votes
1
answer
160
views
Stone-topological/profinite equivalence for quandles
A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$:
(Q1) ...
20
votes
2
answers
1k
views
Several questions about Gauss's mathematical conception of braids
I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
9
votes
1
answer
308
views
A topological characterisation of a.e. continuity
We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
1
vote
0
answers
46
views
Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?
Below we use Bochner measurability and Bochner integral. Let
$(Y, d)$ be a separable metric space,
$\mathcal B$ Borel $\sigma$-algebra of $Y$,
$\nu$ a $\sigma$-finite Borel measure on $Y$,
$(Y, \...
2
votes
1
answer
120
views
Homeomorphisms of the projective cover of the Cantor set
Let $M$ be the projective cover (e.g, Gleason1958) of the Cantor set $\{-1,1\}^{\mathbb{N}}$. Let $\textrm{homeo}(M)$ denote the group of all homeomorphisms of $M$.
Some of the $\gamma\in\textrm{homeo}...
2
votes
1
answer
173
views
Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product
I've been trying to understand various questions to do with sigma algebras on uncountable product spaces.
Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
0
votes
0
answers
94
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
9
votes
1
answer
449
views
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?
We work in ZFC.
Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$.
A field $E$ is ...
6
votes
2
answers
353
views
Stone-Čech boundary is not extremally disconnected
Recall that a topological space is called extremally disconnected if the closure of every open subset is still open. Every discrete space is of course extremally disconnected, and the standard non-...
1
vote
1
answer
117
views
Extremally disconnected rigid infinite Hausdorff compacta(?)
Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism
$\ h: X\to X\ $ is the identity homeomorphism
$\ h=\mathbb I_X:\ X\to X\...
11
votes
4
answers
2k
views
Early illustrations of topological notions in published work
Cross-posted from HSM: I posted this question a bit more than a week ago but have not gotten any answers at HSM. The only comment on the posting asks if I would accept polyhedral pictures ...
1
vote
1
answer
74
views
Subspaces generated by the orbits of the group of isometries on $C(K)$
Let $X$ be an extremally disconnected compact Hausdorff space with no open points, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D_f$ be the linear span of the functions of the ...
3
votes
1
answer
122
views
Spectrum of continuous functions as a semigroup
Let $X$ be a countable group (with the discrete topology) and let $C_b(X)$ be the ring of continuous bounded functions $X \to \mathbb{R}$. It is known that the maximal spectrum of $C_b(X)$, namely the ...
15
votes
1
answer
424
views
Topology and pcf theory
$\DeclareMathOperator\pcf{pcf}$For simplicity say $\aleph_\omega$ is a strong limit. Let $A=\pcf\{\aleph_n:n\in\omega\}$. Then it follows from basic properties of pcf operation that $X\subseteq A\...
1
vote
2
answers
218
views
A few questions about Tychonoff plank
In the Morita's following article (K. Morita. Some properties of M-spaces), constructing an space $X$ and defining an identification on it.
My first question is how to prove that $S$ is countably ...
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
4
votes
1
answer
149
views
Stone–Čech compactification and an ultrafilter of regular closed sets
$\DeclareMathOperator\cl{cl}\DeclareMathOperator\int{int}$A subset $A$ of a topological space $X$ is called regular closed if $A=\cl
_{X}\int_{X}A$.
The family of all regular closed sets of a ...
0
votes
0
answers
69
views
Example of DS with a dense trajectory in the whole state space
Let $U \subset \mathbb{R}^n$ be an open and connected set. We assume there is a vector field $F \in \mathcal{C}^1(\overline{U})$ giving rise to a DS ($\overline{U}$ denotes the closure)
$$\dot{\mathbf{...
0
votes
1
answer
180
views
Is the space $L^p_{\text{loc}} (\mathbb R^d)$ separable w.r.t. the norm $\|f\|_{\tilde L^p} := \sup_{x \in \mathbb R^d} \|1_{B(x, 1)} f\|_{L^p}$?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let ${\tilde L}^p (\mathbb R^d)$ be the space of ...
4
votes
1
answer
138
views
Is the set of clopen subsets Borel in the Effros Borel space?
Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
36
votes
3
answers
6k
views
In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces.
If it is true that:
In a Topological Space, if there exists a loop that cannot ...
3
votes
1
answer
252
views
Can such a set be simply connected?
$\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) ...
8
votes
2
answers
476
views
Continuous point map for spherical domains
Consider the space $J$ of Jordan domains on the sphere $\textbf{S}^2$, i.e., continuous injective maps from the unit disk into $\textbf{S}^2$ modulo homeomorphisms of the disk. How can one construct a ...
28
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
5
votes
2
answers
571
views
On the boundary of a simply connected set
Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x ...
2
votes
1
answer
276
views
Global control of locally approximating polynomial in Stone-Weierstrass?
Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\...
10
votes
0
answers
310
views
Determinacy coincidence at $\omega_1$: is CH needed?
This is a follow-up to the last part of an old MSE answer of mine. Briefly, an analogue at $\omega_1$ of Steel's equivalence between clopen and open determinacy can be proved assuming $\mathsf{CH}$, ...
4
votes
1
answer
192
views
A problem on Demailly's proof of finiteness theorem of elliptic differential operator
I am reading Demailly's notes on pseudodifferential operators on manifolds. And I cannot understand a statement he had made when he tried to prove that the image of an elliptic differential operator ...
3
votes
1
answer
330
views
Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis?
Let me start with the following
Illustration: Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the ...
0
votes
0
answers
58
views
A generalization of relative interior?
In an infinite-dimension space, the relative interior of a non-empty convex set may be empty. I was wondering whether there is a concept (as a generalization of relative interior) with the following ...
3
votes
0
answers
62
views
Smooth Hamiltonian diffeomorphisms form a Baire space
Let $S$ be a closed surface equipped with an area form $\omega$. In Corollary 1.2 of this paper, Asaoka and Irie demonstrated that Hamiltonian diffeomorphisms which have a dense set of periodic points ...
0
votes
1
answer
92
views
A question about the Stone-Čech compactification and ultrafilter
Let $X$ be a Tychonoff space and let $\beta X$ is the Stone-Čech
compactification of $X$. Assume $f:X\longrightarrow \mathbb{R}$ is a bounded function. Then there exists a function $f^{\beta }:\beta X\...
5
votes
1
answer
200
views
Topological property of convergent sequences being eventually constant
Is there a name in the literature for the topological property that all convergent sequences are eventually constant?
This property seems to occur with some frequency and it would be nice to have a ...
0
votes
0
answers
40
views
T. Isiwata's "T. Isiwata. d-, d*-maps and cb*-spaces."
I need T. Isiwata's article T. Isiwata. d-, d-maps and cb-spaces. Bull. Tokyo. Gakugei Univ. Ser. IV, 29, 1977.
Does anyone have it?
https://mathscinet.ams.org/mathscinet/article?mr=0454902
https://u-...
33
votes
6
answers
2k
views
Is there a topology on growth rates of functions?
I've often idly wondered one can say about the collection of "growth rates". By growth rate, let's say we mean an equivalence class of functions $(0,\infty) \to (0,\infty)$, where two ...
0
votes
0
answers
64
views
T. Hanaoka's "Note on c-realcompact spaces and mappings"
I need T. Hanaoka's article Note on c-realcompact spaces and mappings, Memoirs of the Osaka Kyoiku Univ., Ser. Ill, 26 (1977), 55-58.
Can anyone find it for me?
http://ir.lib.osaka-kyoiku.ac.jp/dspace/...
2
votes
3
answers
1k
views
Countable atomless boolean algebra covered by a larger boolean algebra
Suppose $Q$ is an atomless countable boolean algebra, and $B$ is an arbitrary atomless boolean algebra. $Q$ is unique modulo isomorphisms. There is a subalgebra in $B$ that is isomorphic to $Q$. There ...
2
votes
1
answer
298
views
Measurability of Markov kernel wrt the Borel $\sigma$-algebra generated by the weak topology
Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}...
25
votes
6
answers
2k
views
Are there infinitely many "generalized triangle vertices"?
Briefly, I'd like to know whether there are infinitely many "generalized triangle centers" which - like the orthocenter - are indistinguishable from a vertex of the original triangle. This ...