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Questions tagged [gn.general-topology]

Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

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2answers
942 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 ...
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votes
2answers
439 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 ...
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4answers
3k views

Is every norm in R^n a continuous function?

Is every norm in R^n a continuous function?
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votes
2answers
216 views
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1answer
157 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 $M^p≠O^p$?
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votes
1answer
260 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 ...
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votes
1answer
65 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 ${\cal C}$ ...
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votes
1answer
142 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 ...
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votes
2answers
546 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^{\...
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votes
1answer
155 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 ...
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votes
1answer
60 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} $ ...
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votes
1answer
81 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 ...
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votes
1answer
215 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 ...
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votes
1answer
73 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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votes
1answer
69 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, ...
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votes
1answer
261 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 ...
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votes
1answer
473 views

Fuzzy topology : references [closed]

Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
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0answers
69 views

Is there a standard definition for this topology setup?

There are two complete metric spaces $(A,d_A)$ and $(B,d_B)$, also $B \subset A$, so there is a third induced metric space $(B,d_A)$. There is a continuous and onto function $e:A\to B$. For any $b \in ...
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votes
1answer
247 views

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 ...
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votes
2answers
788 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 ...
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votes
2answers
101 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 $f:X\to (X\times X)$?
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votes
2answers
152 views

A countable polish space must 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 ...
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votes
1answer
137 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 ...
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votes
1answer
177 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 $x$...
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votes
1answer
105 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?
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votes
1answer
292 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 ...
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votes
1answer
473 views

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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0answers
234 views

Why is the notion of compactness so powerful? [closed]

According to you, what are the deep reasons, at a very fundamental level, that makes the notion of compactness so useful and ubiquitous throughout modern mathematics: theory of compact groups, compact ...
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votes
3answers
418 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 ...
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votes
1answer
411 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 ...
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votes
1answer
172 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
2answers
185 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\...
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votes
1answer
268 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
2answers
300 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 ...
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votes
1answer
202 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
4answers
618 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....
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votes
1answer
209 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 ...
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votes
2answers
960 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 ...
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votes
1answer
451 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
2answers
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 ...
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votes
1answer
377 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
1answer
467 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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votes
1answer
5k views

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 ...
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votes
1answer
139 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 ...
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votes
1answer
44 views

Two notions of boundedness in metrizable topological vector space

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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votes
2answers
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 ...
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votes
2answers
1k 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} =...
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votes
1answer
1k 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 ...
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votes
1answer
1k views

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 ...