# Questions tagged [gn.general-topology]

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

2,950 questions
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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### 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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### Is every norm in R^n a continuous function?

Is every norm in R^n a continuous function?
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### In a metrizable order topology, is the order relation compatible with the metric? [closed]

Does $x \le y \le z$ imply $d(x, y) \le d(x, z)$?!
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### 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 $M^p≠O^p$?
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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### 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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### 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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### 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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### 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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### 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 ...
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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### 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 $f:X\to (X\times X)$?
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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### 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 ...
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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### 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?
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 ...
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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### 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 ...
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 ...
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 ...
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-...
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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### 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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### 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 ...
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.
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....
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 ...
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 ...
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 ...
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 ...
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 ...
467 views

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 $G_\... 1answer 5k views ### How to transform a plane into a sphere? [SOLVED] [closed] Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ... 1answer 139 views ### Topological spaces without retracts [closed] Is there a way to see whether a topological space$\Omega$does not allow retractions$r: \Omega \mapsto B$, with$B$a given subspace of$\Omega$? In other words: when is a space not retractable ... 1answer 44 views ### Two notions of boundedness in metrizable topological vector space In a metrizable topological vector space X with the metric d, a subset A is said to be bounded if it can be absorbed by any neighbourhood of 0 and a subset A is said to be d-bounded if its diameter ... 2answers 1k views ### Special infinitary relations and ultrafilters (This problem appeared in face of me trying to generalize my theory of (binary) funcoids to the theory of$n$-ary funcoids (I call them "multifuncoids") for arbitrary$n$.) Let$I$is some indexing ... 2answers 1k views ### Filters and intersection of two binary relations Let$\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} =...
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 ...