All Questions
5,184 questions
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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 $x$...
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votes
1
answer
389
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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 ...
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votes
1
answer
111
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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?
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1
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395
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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 ...
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1
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476
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Countable open subgroup
In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
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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
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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
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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
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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
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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-...
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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
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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 ...
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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
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1
answer
234
views
A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
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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.
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votes
4
answers
678
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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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
1
answer
224
views
Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? [closed]
Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there ...
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votes
2
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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 : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...
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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 ...
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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_\...
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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 ...
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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
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votes
1
answer
313
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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 ...
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1
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483
views
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 ...
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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:\...
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votes
2
answers
1k
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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 ...
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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 ...
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1
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2k
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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} =...
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votes
1
answer
2k
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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 ...
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votes
1
answer
2k
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Direct product of filters
Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.
I will denote the principal filter ...
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