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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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 ...

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