All Questions
Tagged with general-topology or gn.general-topology
4,601 questions
0
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85
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Right split for homomorphism onto $S_\infty$
Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...
- 2,882
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votes
0
answers
106
views
Fixed point shape property
Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property.
Here is the definition of f.p.s.p.("map" means ...
- 7,500
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0
answers
173
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Minimum regular open set containing a given set in a T0 Alexandrov topological space
What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be first-...
- 1
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0
answers
331
views
Idempotent ideal in ring of continuous functions
Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be be an idempotent ideal?
- 1
0
votes
0
answers
238
views
Pro-constructible subset of scheme intersects very dense subsets?
Let $X$ be a scheme, let $D$ be a very dense subset of $X$ and let $Y$ be a pro-constructible subset of $X$. Is it true that $Y \cap D \neq \emptyset$?
If $Y$ is just constructible, this is true.
...
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0
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114
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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1
answer
284
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Creating topological spaces with portals [closed]
I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension.
I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
- 19
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153
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extension of function in an abstract metric space
my question is the following.(Maybe my title is not quite proper for this question):
Let $(E,d)$ be a Polish space (or a separable metric space), let $\xi: E\to R_+$ be a Lipschitz function. Now set $...
- 1,835
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0
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161
views
question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
- 1,835
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84
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Can a "weak" topological space be a Moore space?
Let B be a reflexive and infinite dimensional real Banach space-which could be Hilbert space l^2- and let B be endowed with the weak topology. Although this topology is regular and Hausdorff, it is ...
- 6,215
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208
views
A noncommutative analogy of the tube lemma
Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to \mathbb{C}\...
- 356
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answers
109
views
Characterising singular homology among a more general class of cosimplicial spaces
Is there a way to characterise (up to isomorphism) the cosimplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...
- 1,105
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votes
0
answers
128
views
minimal (strongly) KC
If P is a topological property, then a space (X, τ) is said to be minimal P (respectively, maximal) if (X,τ) has property P but no topology on X which is strictly smaller (respectively, strictly ...
- 21
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94
views
Extending coverings over dense subsets
Let $X$ be a metric space with $D⊆X$ a dense subset.
If there is a covering for $D$, under which conditions on the covering is it possible to guarantee that the covering also covers $X$?
For a ...
- 1
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0
answers
345
views
Is $f$ continuous?
The question is also posted here.
The paper is Mizokami : On characterizations of spaces with $G_\delta$-diagonals
See its Theorem 1, also you can see the picture . http://picpaste.com/a-eaiF4d3t.bmp.
...
- 654
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0
answers
148
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Only finitely many fundamental groups in $M(n,k,v,D)$?
Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...
- 689
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answers
196
views
measurable function on a locally compact space for a regular measure
A well known classical fact is that a Lebesgue measurable function on Euclidean space is almost everywhere equal to a Baire class 2 function. A relatively modern reference for this fact is van Rooij -...
- 1,704
0
votes
1
answer
304
views
a questions about the sums of intersections of maximal ideals
why the z-ideals in C(X) are basically the sums of intersections of maximal ideals?
- 1
0
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0
answers
218
views
When is $\{ x \ge 0 | f(x) \le 0\}$ path-connected?
I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x \ge 0 | f(x) \le 0 \}$ is path-connected. We can assume that $f$ is continuous and concave.
...
- 693
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635
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Do homotopic non-intersecting simple closed curves separate the surface?
Let $C_1$ and $C_2$ be two simple closed curves on an orientable compact surface $S$, such that:
They are homotopic to each other.
They are set-theoretically disjoint.
Is $S\setminus(C_1 \cup C_2)$ ...
user13946
0
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0
answers
559
views
Visualizing self-homeomorphism of a cylinder over a torus
A cylinder over a torus is by definition $S^1 \times S^1 \times I$ , here $I=[0,1]$.
One way to visualize it is to thick a torus in $\mathbb{R}^3$. ( $S^1 \times I$ is an annulus, and revolve it (...
- 93
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0
answers
850
views
Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$
I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck:
We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
- 1
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
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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
366
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
167
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 $\mathcal{X}$ ...
- 349
-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
501
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