All Questions
Tagged with gn.general-topology reference-request
325 questions
7
votes
1
answer
1k
views
Reference request: norm topology vs. probabilist's weak topology on measures
Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
- 710
3
votes
1
answer
155
views
A pair of spaces equivalent to a pair of CW-complexes
Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that
$Z-A$ is homeomorhic to $X-Y$ and
$Z/A$ homeomorphic to $X/Y$ and
The closure of $Z-A$ ...
- 855
1
vote
2
answers
195
views
Reference request: lower sets of a preorder form a lattice
Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...
- 695
2
votes
0
answers
126
views
Homeomorphic extension to totally disconnected sets
Dear Mathoverflow Community,
I am looking for a reference for the following topological fact:
Fact
Let $E$ and $F$ be two totally disconnected compact subsets of the plane (can assume perfect if ...
- 2,154
4
votes
0
answers
105
views
Borel selections of usco maps on metrizable compacta
The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
- 41.9k
3
votes
1
answer
143
views
A reference for a (folklore?) characterization of K-analytic spaces
I am writing a paper on K-analytic spaces and need the following known characterization.
Theorem. For a regular topological space $X$ the following conditions are equivalent:
(1) $X$ is a continuous ...
- 41.9k
2
votes
2
answers
185
views
Piecewise-metrizability problems from Willard's Topology
Maybe someone familiar with Willard's textbook can help me out. Problem section 23G on pg. 174 is titled piecewise metrizability. The first problem is:
If a Tychonoff space $X$ is the union of ...
- 553
8
votes
2
answers
577
views
A reference to a well-known characterization of scattered compact spaces
It is well-known that a compact Hausdorff $X$ space is scattered if and only if admits no continuous maps onto the unit interval $[0,1]$.
Surprisingly, but I cannot find a good reference to this well-...
- 41.9k
2
votes
0
answers
120
views
Two small uncountable cardinals related to Q-sets
A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$.
Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
- 41.9k
0
votes
1
answer
152
views
Reference request: Baire class 2 functions
There are many articles on Baire 1 functions, but not many on Baire 2 and above. Where can I find a nice comprehensive survey of them?
- 2,069
11
votes
1
answer
355
views
Name for topological spaces where "every point has a local base wellordered by reverse inclusion"?
There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base.
Is there a similar name for a space where "every point has a ...
- 775
1
vote
1
answer
1k
views
A new generalisation of dimension? part 2
I worked this theory : A new generalization of the dimension?
I have a theorem about dimensions which is more general and simple than for matroids.
Definition 1: A structure $S$, is a pair $(X, \...
- 4,074
7
votes
0
answers
119
views
The automorphism group of the fibered cylinder
My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
- 41.9k
3
votes
1
answer
95
views
sequences of iterated orthogonals (lifting property) in a category
I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.
For example, several iterated orthogonals of $ \emptyset\...
- 99
3
votes
0
answers
102
views
Find a certain triangulation subordinate to a given covering of a manifold
Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
- 2,789
6
votes
0
answers
563
views
Grothendieck letter to Jun-Ichi Yamashita on tame topology
I am looking for Grothendieck writings on tame topology:
a manuscript on tame topology mentioned by Scharlau; a letter to Jun-Ichi Yamashita; a letter to Z.Mebkhout.
I am also interested in ...
- 99
4
votes
1
answer
153
views
The homological negligibility of certain subsets in compact manifolds
Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).
I need a reference to the following facts (which I believe are true at least in dimension $n=3$):
Fact 1. For every ...
- 41.9k
2
votes
1
answer
266
views
Approximate selection for finite-valued upper hemicontinuous/semicontinuous maps?
I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite ...
- 85
6
votes
1
answer
186
views
Reference request: A collection of topologies on $\mathbb{N}$ formed via series
First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean ...
- 6,546
4
votes
1
answer
217
views
Partitions of unity in constructive mathematics
Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics?
In particular, under what circumstances can we construct a partition of unity subordinate to ...
- 791
5
votes
2
answers
1k
views
An example of an open discontinuous function
Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the ...
- 1,897
5
votes
0
answers
330
views
The second dual of $C(X)$ with the compact-open topology
Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
- 233
1
vote
2
answers
223
views
Is $C_b(Q,E)$ linearly isometrically isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone–Čech compactification of $Q$?
Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space.
Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ and $C_b(Q,E)$ be the collection of all $E$-valued ...
- 623
9
votes
0
answers
569
views
A standard name for a function satisfying the intermediate value theorem?
Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...
- 41.9k
8
votes
0
answers
110
views
Connected component optimization
For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...
- 623
3
votes
1
answer
578
views
Existence of a discrete subset
Let $X$ be a topological space. $Y$ is a discrete subset of $X$ if it has a discrete topology induced by the topology of $X$. This is equivalent to the fact that for every $y\in Y$ there is an open $U\...
- 5,529
9
votes
1
answer
322
views
What is the (genuine) name for the Gutik hedgehog?
Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named Gutik's hedgehog. It is homeomorphic to the space
$$H:=\{(0,0)\}\cup\{(\tfrac1n,0):...
- 41.9k
13
votes
1
answer
570
views
Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects
These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
- 18.4k
3
votes
1
answer
436
views
Stone topological Boolean algebras
I am looking for an initial reference for a theorem which is known, namely:
Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
- 774
7
votes
0
answers
305
views
Generalizing Gromov Hausdorff distance using Vietoris topology
There are two notions of convergence of a sequence of metric space. One is by the Gromov Hausdorff distance for compact metric spaces, another one is the pointed Gromov Hausdorff convergence for ...
- 1,630
9
votes
2
answers
1k
views
A definition of topology using monads (a.k.a. halos)
In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all ...
- 6,353
7
votes
2
answers
608
views
What is the name for a set endowed with a Lipschitz structure?
I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
- 41.9k
8
votes
1
answer
276
views
Almost convex combinations in $\mathbb R^n$
Working on some problems in the $C_p$-theory I discovered the following simple but amazing
Fact. For any subset $A\subset \mathbb R^n$, non-zero vector $a\in \bar A\subset\mathbb R^n$ and $\...
- 41.9k
1
vote
1
answer
179
views
Is there any upper bound on the LS-category of open $n$-dimensional submanifolds of $\mathbb{R}^n$?
Suppose $X\subseteq\mathbb{R}^n$ is a connected open bounded $n$-dimensional submanifold.
1) I wonder if for the class of such spaces there is any upper bound on the LS-category of $X$?
2) Is it ...
- 3,173
5
votes
1
answer
381
views
Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$
I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...
- 734
3
votes
0
answers
56
views
Name for a special kind of neighborhood assignment or for the existence thereof
Lets say temporarily that a topological space $(X,\tau)$ is weird if there is a function $\varphi:X \to \tau$ such that for all $x \in X$:
$x\in\varphi(x)$,
$\{y\in X: x \in \varphi(y)\}$ is finite.
...
- 11.5k
7
votes
3
answers
310
views
Non-metric topological continua
What important results hold for non-metric continua, or where can I find a survey of such results?
There are three definitions of a continuum around: a non-empty topological space that is
(1) ...
- 504
2
votes
0
answers
65
views
Splitting of ordinals of oscillation ranks of a Baire $1$ function
Denny and Tang proved that
Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$
Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
- 623
0
votes
1
answer
377
views
How to prove that there does not exist any plane isotopy from the logarithmic spiral onto the real line? [closed]
Questions.
EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a strong a-forteriori-reason, kindly ...
- 6,051
10
votes
0
answers
441
views
A new $\ell_p$-metric on the hyperspace of finite sets?
Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n:=...
- 41.9k
12
votes
4
answers
1k
views
What was Burroni's sketch for topological spaces?
In a 1981 talk, René Guitart cites Albert Burroni as having given "A first interesting example of a mixed sketch...for the category of topological spaces" in 1970. This was apparently done in Burroni'...
- 3,411
4
votes
2
answers
149
views
Which necklaces require maximal cuts?
Given an unclasped necklace with $d$ types of beads and $p$ people it is well known we can fairly divide the necklace with at most $d(p-1)$ cuts. A fair division means that each person is given the ...
- 143
3
votes
0
answers
78
views
Nowhere dense covering number of a connected $T_2$ space
This is a generalization of an older question.
If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a ...
- 48.9k
12
votes
1
answer
316
views
A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube
I am looking for a reference of the following (true) fact:
Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
- 41.9k
17
votes
1
answer
988
views
Can two-point sets be Borel?
Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.
I wonder if the following question of ...
- 32.1k
16
votes
2
answers
820
views
Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
- 45k
9
votes
4
answers
1k
views
When $X \times Y \cong X \times Z$ implies $Y \cong Z$ (in the category of finite topological spaces)
The title has it all. I'm looking for a reference to the following:
Q. Let $X, Y, Z$ be finite, non-empty (topological) spaces. When does $X \times Y \cong X \times Z$ imply $Y \cong Z$ (in the ...
- 10.5k
11
votes
1
answer
441
views
Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 8,071
9
votes
1
answer
2k
views
When will the supporting hyperplane of a convex set coincide with the tangent?
Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I ...
- 8,071
10
votes
1
answer
695
views
Topology from the viewpoint of the filter endofunctor
Question. Are there any references that develop general topology from the viewpoint of a functor $$\Phi : \mathbf{Rel} \rightarrow \mathbf{Rel}$$ that assigns to every set $X$ the set $\Phi(X)$ of ...
- 3,793