All Questions
5,184 questions
0
votes
1
answer
194
views
Difference between a partial selector and a selector...
In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem:
There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set.
The proof is as follows:
...
- 463
0
votes
0
answers
189
views
On Birman-Wenzlyfying the B2 spider
Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying
the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible
S matrices (paper pending) - it's ...
- 4,829
0
votes
0
answers
365
views
Finding paths in a path connected space
I'm looking for such literature as exists relevant to the following problem.
Problem Given a compact, path-connected region $E$ on the plane and a positive constant $r$. Find (if possible) a path ...
- 627
0
votes
1
answer
271
views
Numbers associated with boundaries of manifolds
I don't know what name if any is attached to the numbers I'm about to describe.
For a line segment, [a,b]
the number is 1 if for any k in (a,b)
and 2 if k=a or k=b.
For a square, [a,b] ...
- 379
-1
votes
3
answers
523
views
Metric properties for $d:X\times X\times\dotsb X\rightarrow\mathbb R$ [closed]
Let us define $d:X^n\rightarrow\mathbb R$. How can we define metric properties such as symmetry, triangle inequality equivalent property etc for such a function?
- 243
-1
votes
2
answers
409
views
$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
- 263
-1
votes
1
answer
96
views
Limiting points of elementary set
I consider the following set
$$A:=\left\{ \frac{3mn}{2(m^2+mn+n^2)}; m,n \in \mathbb Z; \text{ and }m,n \text{ are not both zero}\right\}$$
Is it possible to identify the closure of $A$ in the reals?
- 124
-1
votes
1
answer
367
views
When is any convergence sequence is stationary?
Is there any characterization for a topological space under which every convergent sequence is stationary? (e.g. discrete topology)
-1
votes
1
answer
168
views
Space of distributions on $[0,1]^2$: weakly compact or not?
Let $X_1,X_2$ be distributions on $[0,1]$ and let $X=(X_1,X_2)$ be the joint distribution of $X_1,X_2$. Let $\mathcal{X}$ be the set of all such joint distribution $X$.
Question 1: Does $\mathcal{X}$ ...
- 359
-1
votes
2
answers
260
views
Function space and contractibility
$\DeclareMathOperator\map{map}$I have the following question:
Let $X$ and $Y$ be topological spaces. Let $\map(X,Y)$ denote the space of non-constant continuous functions from $X$ to $Y$. Suppose ...
- 315
-1
votes
1
answer
1k
views
Graph of function, continuous projection [closed]
$X$ and $Y$ are topological spaces. $f:X\rightarrow Y$ a map (we don't suppose that $f$ is continuous). Consider
$A=\{(x,f(x))\in X\times Y| x\in X\}$. is $\pi: A\rightarrow X$, $$(x,f(x))\mapsto x$$...
- 71
-1
votes
1
answer
122
views
Injective choice function for non-separable $T_2$-spaces
For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ be the collection of all subsets of $X$ of cardinality $\kappa$.
I was looking for $T_2$-spaces $(X,\tau)$ with the property that
$(P)$ ...
- 48.9k
-1
votes
1
answer
517
views
On ultraproducts of topological spaces
Intuitively, I understand the construction of the hyperreals by ultraproducts as a process of turning the limit operation into an algebraic object. More precisely, to check the existence of the limit $...
- 563
-1
votes
1
answer
267
views
When is the orbit space of a manifold still a manifold of the same dimension?
$\mathbf{Question}$. Let us assume that $M^n$ is a topological manifold of dimension n, with a group action $\Gamma$, which acts discontinuously and freely on $M^n$. Is the orbit space $M^n / \Gamma$ ...
- 33
-1
votes
2
answers
466
views
Union of proximally connected sets
Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Is the following true? (I need a proof or a counter-example.)
Conjecture If S ...
- 765
-1
votes
1
answer
231
views
The set of prime numbers as a subspace of the Cantor set
We define an embedding of the set of prim numbers into the Cantor set as follows:
First we recall that the cantor set $\mathcal{C}$ is homeomorphic to $(\mathbb{Z}/10\mathbb{Z})^\omega $ since the ...
- 356
-1
votes
1
answer
116
views
Continuous surjection of $\mathbb{R}^{n-1}$ onto the interior of the $n$-simplex with continuous right inverse
Let $n$ be a positive integer. Clearly $\mathbb{R}^{n-1}$ and the interior of the $n$-simplex $
\delta_n := \{x \in [0,1]^n:\,\Sigma_k x_n =1, (\forall i)\,x_i>0\}
$ are homeomorphic. What I'm ...
- 5,405
-1
votes
2
answers
502
views
In a metrizable order topology, is the order relation compatible with the metric? [closed]
Does $x \le y \le z$ imply $d(x, y) \le d(x, z)$?!
-1
votes
1
answer
187
views
Existence of a special type of maximal ideal in $C(X)$:
Does there exist any maximal ideal $M^p$ in $C(X)$ (the ring of continuous functions on a topological space $X$) such that each element of $M^p$ is a divisor of zero but $M^p≠O^p$?
- 5
-1
votes
1
answer
346
views
An infinite set in a compact space
Let $X$ be a topological space. Is there any characterization for the property that says "for every infinit subset $A$ of $X$ there exists $a\in A$ such that if $f$ be an arbitrary real continuous ...
-1
votes
2
answers
325
views
Corresponding between prime ideals in $C(X)$ and $C^*(X)$
we know that every maximal ideal in $C(X)$ is in this form:
$$M^p=\left\{\,f \in C^*(x):\ p\in cl_{\beta X} Z\left(f\right)\,\right\}$$
and every maximal ideal in $C^*(X)$ is
$$M^{*p}=\left\{\,f\...
- 35
-1
votes
1
answer
85
views
Intersection of complements of connected components (2)
Let $(X,d)$ be a non-compact, complete metric space and $K\subseteq X$ compact. Pick $x^* \in X\setminus K$.
Let $E$ be the connected component of $X\setminus K$ that contains $x^*$. Let ${\cal C}$ ...
- 48.9k
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
-1
votes
2
answers
1k
views
The boundary of this set is piecewise smooth? [closed]
Consider a sequence of open sets in $R^n$: $\Omega_1 \supset \Omega_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean $C^{\...
- 269
-1
votes
1
answer
115
views
Is this function on the Cantor set continuous? [closed]
Let $S = \displaystyle \prod_{n \ge 1} \{ 0, 1\}$ be the set of binary sequences, so $S$ with the product topology is homeomorphic to the Cantor set. Endow $\mathbb{Z}_{\ge 1} \cup \{ \infty \}$ with ...
- 233
-1
votes
1
answer
153
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 ...
- 173
-1
votes
1
answer
98
views
Topological connected eccentrics, not homeomorphic to commutative Lie groups
An eccentric is a universal algebra $\ (X\ \sigma\ \lambda\ \rho)\ $ such that operations
$\ \sigma\ \lambda\ \rho\,:\,X\times X\to X\ $ satisfy:
$\quad \forall_{x\ y\,\in X}\quad \lambda(\sigma(x\ y)...
- 4,786
-1
votes
1
answer
256
views
Injectivity of a locally strictly expanding map on a compact space
Prove that any locally strictly expanding map on an infinite compact metric space is non-injective.
- 189
-1
votes
1
answer
80
views
Minimal covering sets of continuous endomorphisms
For any topological space $(X,\tau)$, let $\text{End}(X)$ denote the set of continuous functions $f:X\to X$. We say that ${\cal C}\subseteq \text{End}(X)$ covers $\text{End}(X)$ if for every $f\in \...
- 48.9k
-1
votes
1
answer
81
views
Closed on generic set implies closed set whole set [closed]
Assume that $f:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is a continuous on a set $A$. Let $B \subset A$ be a generic set in $\mathbb{R}^{2}$ i.e, the countable intersection of the open and dense ...
- 1,043
-1
votes
1
answer
339
views
A condition for Artinian topological spaces [closed]
A topological space $X$ is called Artinian if the descending chain condition holds for open subsets of $X$. If the descending chain condition holds for open basis subsets of a Hausdorff space $X$ with ...
- 13
-1
votes
1
answer
73
views
existence of continuous functions with values in the fiber of a closed bundle
Let $ A \subseteq \mathbf{R}^{n} $ be a closed set and let $ B $ be a closed unit normal bundle over $ A $ ( that means for every $ a \in A $ we have closed subset $ B_{a} \subseteq \mathbf{S}^{n-1} $ ...
- 541
-1
votes
1
answer
88
views
Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$
Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...
- 48.9k
-1
votes
1
answer
278
views
Decomposition space of $\mathbb{C}$ by concentric circles [closed]
What are the topological properties of the quotient space $X$ obtained from $\mathbb{C}$ by identifying points of the same modulus? I.e., the space $X=\mathbb{C}/E$ where $E$ is the equivalence ...
- 1,704
-1
votes
1
answer
81
views
extension of a continuous function [closed]
Please is it true that if $f:K\to \mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of $K$?
...
- 1
-1
votes
1
answer
110
views
Variety of commutative semi group [closed]
V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.
- 155
-1
votes
1
answer
75
views
Finiteness of "novel variance" from a kernel on a compact space [closed]
Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
- 8,512
-1
votes
1
answer
669
views
Stone-Cech compatification and ultrafilter [closed]
I have been studding about compatification of a topological space $X$. But I have low understanding about the Stone-Cech compatification, specially construction of the Stone-Cech compatification on ...
- 147
-1
votes
1
answer
416
views
the space of maximal ideals in C(X) and C*(X) [closed]
Let $C(X)$ be the continous function ring and $C*(X)$ be the bounded continous function ring.$Max C(X)$ consisting of all maximal ideals in $C(X)$.
Question:why $Max C(X)$ and $Max C*(X)$ are compact ...
- 21
-1
votes
1
answer
542
views
Fuzzy topology : references [closed]
Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
- 11
-1
votes
1
answer
162
views
A topological space whose closed subsets are locally connected
Let $X$ be a compact $T_0$ topological space such that every closed subset of $X$ is locally connected. Is there any characterization for such a space? I guess $X$ is Noetherian, but I cannot prove ...
-1
votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362
-2
votes
2
answers
931
views
Can topologies induce a metric?
Let {X,T} be a topology, T the set of open subsets of X.
Definition: Three points x, y, z of X are in relation N (Nxyz, read "x is nearer to y than to z") iff
there is a basis B of T and b in B ...
- 9,720
-2
votes
2
answers
124
views
Example of connected Hausdorff space $X$ and surjective continous map $f:X\to X\times X$ [closed]
What is an example of a connected Hausdorff space $X$ with $|X|>1$ and a surjective continous map $f:X\to (X\times X)$?
- 48.9k
-2
votes
2
answers
674
views
Must a countable Polish space be discrete? [closed]
I am looking for an elegant proof of the fact that a countable metric space is complete iff its underlying topology is discrete.
It is easy to see that a discrete space is complete because its ...
- 2,655
-2
votes
1
answer
314
views
configuration space and iterated loop space
Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...
- 1,990
-2
votes
1
answer
458
views
some trouble over the cardinality of the cantor set(middle one-thirds) [closed]
firstly i thank you for taking interest in my post but i am new here so if i have made some mistakes or done something which is out of place please point out.my problem is-
we know that the cantor ...
- 19
-2
votes
1
answer
131
views
$G$- space is locally compact [closed]
Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
- 451
-2
votes
2
answers
954
views
Three modifications of connectedness
This question arose in my research of generalized connectedness (see this draft article for the overall idea, but beware that the draft is yet too preliminary and unreadable, however I hope you can ...
-2
votes
1
answer
780
views
commutative monoids have binary products? [closed]
Does the category CMonoid of commutative monoids have binary products?
thanks
- 1