All Questions
Tagged with general-topology or gn.general-topology
4,601 questions
-1
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1
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80
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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
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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 $K$?
...
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-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
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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
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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
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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
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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
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
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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
1k
views
Component and quasi-component
Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$...
- 1
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votes
1
answer
389
views
Bounded metric spaces with non-surjective self-isometry
A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
- 48.9k
-2
votes
1
answer
111
views
Is this space discrete? [closed]
Let X be a Tychonoff space such that for any closed set A there exist a continuous function f: X to R such that A=cl(X-Z(f)). Is this space X discrete?
- 5
-2
votes
1
answer
395
views
non-trivial convergent sequence [duplicate]
I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...
- 147
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votes
1
answer
476
views
Countable open subgroup
In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
- 7
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1
answer
315
views
Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
- 4,074
-3
votes
2
answers
7k
views
Continuous map from $\mathbb R^2$ to $\mathbb R$? [closed]
There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$?
I guess what I mean is ...
-3
votes
3
answers
753
views
Riemann Mapping Theorem in Higher Dimensions for Continuous funcions [closed]
Is there any analogue for Riemann Mapping Theorem(!) in higher dimensions?
Or a much simpler question, is it true that every open subset of $\mathbb{R}^3$ with zero homology in dimensions 1 and 2 is ...
- 615
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votes
1
answer
361
views
Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\DeclareMathOperator\CM{CM}$
I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
- 5,405
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votes
2
answers
1k
views
Finite versus infinite on non-Hausdorff topologies [closed]
Question: Does there exist some real-valued function $f(x)$ where $f: \mathbb{R} \to \mathbb{R}$, for which $\lim_{x \to \infty}$ converges on a non-Hausdorff topology but does not converge on a ...
- 283
-3
votes
1
answer
125
views
Basis of Euclidean topology on $\mathbb{R}$ such that no element is contained in another [closed]
What is an example of a topological base ${\cal B}$ for $\mathbb{R}$ with the Euclidean topology such that for every $B_1\neq B_2 \in {\cal B}$ we have $B_1\not\subseteq B_2$?
- 48.9k
-3
votes
1
answer
191
views
The usual topologies [closed]
My questions are :
Why do we commonly use certain usual topologies rather than others ? For example the usual topology on the real numbers, the topology of
uniform convergence, the compact-...
- 129
-3
votes
1
answer
211
views
Can a Polish space have two different topologies?
Let $X$ be a Polish space with the compatible metric being $d_1$. So $(X,d_1)$ is a separable complete metric space, and the topology is generated by $d_1$.
Can there be a metric $d_2$ such that $(X,...
- 291
-3
votes
1
answer
330
views
Loop space of manifold [closed]
Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...
- 189
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votes
2
answers
314
views
Dispensing with the notion of infinity for the sake of coverings [closed]
Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but ...
- 117
-3
votes
1
answer
230
views
Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]
Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)?
If so, please show me how to construct it.
- 3
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votes
4
answers
677
views
What is the max number of points in R^3, interconnected by generic curves?
The largest complete graph that embeds in 2 dimensions is $K_4$, while the largest complete graph that embeds in 3 dimensions is $K_{\infty}$, right? However, I don't know any constructive proof of it....
- 555
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votes
1
answer
328
views
Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]
Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?
NOTE: PLEASE avoid the ...
- 612
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votes
2
answers
405
views
Do these irrationals exist?
An irrational $a$ verifies : $\{a\times n+k;(n,k)\in\mathbb Z^2 \}$ is dense in $\mathbb R$.
If you take $a$ universe then : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...
- 4,074
-4
votes
1
answer
412
views
A topological groupoid structure on a pair $(X,A)$
Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.
Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding ...
- 356
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votes
1
answer
483
views
Why $z \in \overline{A}$? [closed]
In the Picture blew:
The paper can be downloaded here. Why $z \in \overline{A}$?
Thanks.
A point $x$ of a space $X$ is called $G_\omega$-separated from a subset $Y$ of $X$ if there is a closed $G_\...
- 654
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votes
1
answer
8k
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How to transform a plane into a sphere? [SOLVED] [closed]
Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...
- 555
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votes
1
answer
97
views
Two notions of boundedness in metrizable topological vector space [closed]
In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ...
- 93
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votes
1
answer
177
views
Topological spaces without retracts [closed]
Is there a way to see whether a topological space $\Omega$ does not allow retractions $r: \Omega \mapsto B$, with $B$ a given subspace of $\Omega$ ?
In other words: when is a space not retractable ...
- 4,547
-5
votes
1
answer
313
views
Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]
In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
- 6,974
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1
answer
483
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For every proximity, does there exist a uniformity which generates this proximity?
For every proximity, does there exist a uniformity which generates this proximity?
This question may be generalized for different generalizations of proximities and uniformities. In fact I need it ...
- 765
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votes
1
answer
175
views
Continuous function $f:\mathbb{R}\to\mathbb{R}$ with fixed size finite fibers [closed]
During a business meeting, I was trying to find a continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $|f^{-1}(\{y\})| = 2$ for all $y\in \mathbb{R}$, and after some experimentation I found $$f:\...
- 48.9k
-8
votes
2
answers
1k
views
Special infinitary relations and ultrafilters
(This problem appeared in face of me trying to generalize my theory of (binary) funcoids to the theory of $n$-ary funcoids (I call them "multifuncoids") for arbitrary $n$.)
Let $I$ is some indexing ...
- 765
-8
votes
1
answer
351
views
Are there overwhelmingly more finite monoids than finite spaces? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
- 23
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votes
1
answer
2k
views
Filters and intersection of two binary relations
Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} =...
- 765
-11
votes
1
answer
2k
views
Union of uniformly connected sets
I will call a set uniformly connected regarding some uniform space when it is connected regarding every entourage of this uniform space (entourages are considered as digraphs and it is taken strong ...
- 765