All Questions
Tagged with gn.general-topology reference-request
306
questions
2
votes
1
answer
200
views
Is the Fortissimo space on discrete $\omega_1$ radial?
Let $\omega_1$ have the discrete topology. Its Fortissimo space is $X=\omega_1\cup\{\infty\}$ where neighborhoods of $\infty$ are co-countable.
A space is radial provided for every subset $A$ and ...
- 933
9
votes
1
answer
501
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 289
6
votes
4
answers
473
views
Countable chain condition in topology
A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...
- 359
10
votes
1
answer
351
views
Two dimensional perfect sets
Consider the following family of sets
$$ \begin{align*}
\mathcal{F} = \{X\subseteq [0,1]\times [0,1] \mid \ &X \text{ is closed and }\\& \forall x \in \pi_0 (X) (\{y \in [0,1] \mid (x,y) \in ...
- 2,042
4
votes
0
answers
156
views
When $X$ is homeomorphic to $\mathscr{F}[X]$?
While I was talking to some colleagues, one of them said that there exists a topological space $X$ such that $X$ is uncountable, non-discrete and homeomorphic to $\mathscr{F}[X]$ (the Pixley-Roy ...
- 151
1
vote
0
answers
75
views
Is there a standard name for the following class of functions on non-Hausdorff manifolds?
Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
- 1,958
1
vote
1
answer
75
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 ...
- 933
2
votes
1
answer
115
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}...
- 2,118
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 ...
- 24.5k
2
votes
2
answers
269
views
Topological characterisations of properties of posets
Finite connected partially ordered sets are in bijective correspondence to connected finite topological spaces that satisfy T_0, see for example the Wikipedia article Finite topological space. Here ...
- 25.4k
1
vote
0
answers
107
views
Refinement of an open cover for a simply connected compact subset
Let $U$ denote a simply connected, open subset of the plane, and let $K$ be a simply connected, compact subset of $U$. Can we always find a finite or countable sequence of open disks $(D_n)$ such that:...
- 341
2
votes
0
answers
139
views
Properties of universal fibration
I am trying to read the following paper [1] (Becker, James C.; Gottlieb, Daniel Henry
Coverings of fibrations.
Compositio Math.26(1973)) where the authors mentioned that for any fiber $F$,
there ...
- 179
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 ...
- 6,764
2
votes
1
answer
119
views
Hereditarily locally connected spaces
A topological space is locally connected if every point has a neighborhood basis of connected open subsets.
A property of topological spaces is termed hereditary, subspace-hereditary, if every subset ...
- 774
2
votes
0
answers
150
views
Are there hereditarily square-boxed plane continua?
A plane continuum is a bounded, closed and connected subset of the plane.
A bounding box $B$ for a plane continuum $C$ is
a rectangle $B=[a,b]\times[c,d]$ (including sides and interior)
such that $C$ ...
- 1,345
6
votes
1
answer
443
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 40.2k
3
votes
0
answers
173
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 40.2k
7
votes
0
answers
222
views
A metric characterization of Hilbert spaces
In the Wikipedia paper on Hadamard spaces, it is written that every flat Hadamard space is isometric to a closed convex subset of a Hilbert space. Looking through references provided by this Wikipedia ...
- 40.2k
2
votes
0
answers
87
views
References (and a question) on the "fine" topology of powersets
Recently I've been trying to understand powerset topologies better, and came upon the following reference:
Frank Wattenberg, Topologies on the set of closed subsets. Pacific J. Math. 68(2): 537-551 (...
- 9,727
2
votes
0
answers
64
views
When did derivative algebras first appear?
In the paper "The Algebra of Topology" (Annals of Mathematics, 45, 1944), McKinsey and Tarski proposed derivative algebras (p183) to define the derive set in topology as follows.
Suppose $K$ ...
- 663
3
votes
0
answers
88
views
Constructively valid reference for the soberness of discrete spaces and points of a locale coproduct
I am looking for constructively valid references for the following two related facts:
discrete topological spaces are sober,
the points of a locale coproduct are the disjoint union of the points of ...
- 28.7k
1
vote
1
answer
329
views
Notations for open and closed sets
I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
- 111k
1
vote
1
answer
175
views
When are fixed point sets in $T_1$ spaces always closed?
Let $X$ be a topological space, and say that $X$ satisfies the closed fixed point set property if every continuous self-map $f:X\to X$ has fixed point set $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ ...
- 2,742
16
votes
1
answer
454
views
Where can I learn more about the topology on $\mathbb{R}$ induced by the map $\mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$?
Consider the (continuous, injective, abelian group homomorphism) map $\Phi \colon \mathbb{R} \to \prod_{a>0} (\mathbb{R}/a\mathbb{Z})$ (where the target is given the product topology) taking $x\in \...
- 28.7k
3
votes
0
answers
104
views
"Practical" references on mapping spaces as infinite-dimensional manifolds
I am studying spaces of the form $C^{k}(\mathcal{M},\mathcal{N})$ between manifolds ($k=\infty$ allowed) and I am looking for extensive references, especially analysing their topology and smooth ...
- 833
6
votes
0
answers
188
views
Every Polish space is the image of the Baire space by a continuous and closed map, reference
The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501)
Every Polish space (i.e. every separable ...
- 2,042
2
votes
1
answer
166
views
A stronger version of paracompactness
Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $...
- 665
6
votes
0
answers
246
views
Have we discovered constructions for natural fractional dimensional spheres?
I have been thinking about a couple different problems in fractal geometry (including I one deleted because it was ill posed) and realize they all depend in a fundamental way on the problem of: Can we ...
- 2,233
1
vote
0
answers
81
views
Reference request: rates of weak convergence of Polish space-valued random variables
Let $(E,\mathscr{E})$ be a Polish space $E$ together with its Borel $\sigma$-algebra $\mathscr{E}$. Let $(\Omega, \mathscr{F},P)$ be a probability space and let $(X^{(n)})_{n \in \mathbb{N}}$ be a ...
- 131
2
votes
1
answer
159
views
A characterization of continuity in terms of preservation of connected sets. Where to find the result?
There is a result that if $X$ is a locally connected space and $Y$ is a locally compact Hausdorff space, then a function $f \colon X \to Y$ is continuous if and only if $f$ has a closed graph and for ...
- 149
6
votes
0
answers
128
views
A theorem by R.L. Moore
The following result is due to R.L. Moore.
Let $K\subseteq\mathbb C$ be compact. Suppose that
$K$ is connected,
and that $\mathbb C\setminus K$ is connected.
Then $\partial K$ is connected.
Does ...
- 687
1
vote
0
answers
45
views
Topological rings with a final topology
Given a family of ring homomorphisms $ \phi_i : X \rightarrow Y_i $ where each $ Y_i $ is a topological ring and consider the initial topology on $ X $, i.e. the coarest topology such that each map is ...
user491484
1
vote
1
answer
191
views
Name of a space with both a topology and a metric that are not compatible?
Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...
- 665
2
votes
2
answers
479
views
How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
- 129
3
votes
0
answers
87
views
Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
- 11.9k
4
votes
0
answers
161
views
In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?
I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
- 11.9k
4
votes
0
answers
137
views
Consistency of a strange (choice-wise) set of reals, pt. 2
This is a follow-up on this question. Consider a set $X\subseteq \mathbb{R}$ such that
$X$ is not separable wrt its subspace topology
Every countable family of non-empty pairwise disjoint subsets of $...
- 2,042
2
votes
1
answer
118
views
For which continua $X$ does $X^\mathbb{N}$ have the fixed-point property?
A topological space $X$ has the fixed-point property if any continuous map $f : X \to X$ has a fixed point (i.e., an $x$ such that $f(x) = x$). It's well known that certain topological spaces, such as ...
- 10.1k
2
votes
1
answer
119
views
Is a Boolean algebra with an order continuous topology a measure algebra?
Assume that $B$ is a complete boolean algebra endowed with a Hausdorff topology, with respect to which all operations on $B$ are continuous, $0$ has a base of full sets (recall that $A\subset B$ is ...
- 5,275
7
votes
2
answers
702
views
Consistency of a strange (choice-wise) set of reals
Consider a set $X\subseteq \mathbb{R}$ such that
$X$ is not separable wrt its subspace topology
For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$
In a ...
- 2,042
1
vote
0
answers
39
views
Reference for preimage of boundary of spacefilling curve
Given a continuous map $\gamma$ from $[0,1]$ onto a bounded
contractible subset $S$ of $\mathbb R^2$ such that $S$ contains an open subset of $\mathbb R^2$ which is dense in $S$,
the preimage $\gamma^{...
- 17k
13
votes
2
answers
2k
views
When can we divide continuous functions?
Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$.
What can be said ...
- 5,275
1
vote
0
answers
154
views
Study of the class of functions satisfying null-IVP
$\mathcal{N}_u$ : Class of all uncountable Lebesgue-null set i.e all uncountable sets having Lebesgue outer measure $0$.
Let $f:\Bbb{R}\to \Bbb{R}$ be a function with the following property :
$\...
- 317
0
votes
0
answers
95
views
Reference request: subspace-based generalisation of weak* convergence
Let $V$ be a normed space and $(V_j)_{j\in [0,1]}$ be a family of linear subspaces of $V$ with $V_1$ non-trivial and such that $V_1\subsetneq V_j\subseteq V_i$ whenever $i\leq j$. We write $W:=V'$ for ...
- 411
2
votes
1
answer
133
views
Borel $\sigma$-algebras on paths of bounded variation
Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.
Let further $B\subset C$ be the subspace of $0$-started ...
- 411
2
votes
0
answers
93
views
Concrete topological objects and notions in the category of locales
I have read Peter Johnstone's “The Point of Pointless Topology” and the idea that topological spaces are not quite the right abstraction for topology seems, at least philosophically, rather appealing. ...
- 121
3
votes
1
answer
224
views
What is this property of surjective continuous maps called?
Let $f\colon X\to Y$ be a continuous map between topological spaces, which you can assume to be Hausdorff if you like. Say that $f$ has property $P$ if for every compact subset $L\subseteq Y$, there ...
- 54.6k
4
votes
1
answer
230
views
Product topology from two premetric spaces induced by sum of premetrics?
For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$.
Do ...
- 411
0
votes
2
answers
243
views
Finite sheeted covering of the complement of a finite set in $\mathbb{C}$
For figure "eight" there is a list of finite sheeted covering discussed in Hatcher's book "Algebraic topology". I was thinking about the following question:
Let $S$ be a finite ...
- 1,039
3
votes
1
answer
159
views
Co-analytic $Q$-sets
A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
- 2,042