The point-set-topology tag has no wiki summary.

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### Proving, that closure off set is equal this set iff set is closed [on hold]

I've started intorduction to topology course and I need help with one of the problems:
Let $A \subset(X,T). $ Prove that $cl(A) = A\iff A$ is closed.
It may looks trivial, but I had a little ...

**0**

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**1**answer

66 views

### Can we build a continuous function from “fibers”/preimages defined over a topological base?

I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...

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131 views

### Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer.
Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...

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312 views

### Topological proof of a result in Logic

I proved the result below using logic. My questions:
Can this theorem be proved by purely topological means?
Do you know any theorems that either can be used to prove the same result, or which give ...

**9**

votes

**1**answer

303 views

### continuous images of open intervals

The well-known Hahn-Mazurkiewicz theorem characterizes those nonempty Hausdorff spaces $X$ that admit a continuous surjection $\alpha: [0, 1] \to X$ from the closed unit interval: it is necessary and ...

**0**

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**1**answer

80 views

### Is a weakly separable group always Lindelöf?

By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...

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56 views

### A construction with Hyperspace of continums

Let $X$ be a compact connected metric space. Its hyperspace is denoted by $2^{X}.$ $X$ is considered as a subset of $2^{X}$ via the embedding $x\mapsto \{x\}$. Assume that $f:X\to X$ is a ...

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**1**answer

139 views

### A question about small cardinals related to Michael's Problem

I'm studying some applications of small cardinals related to the Michael's Problem. Recall that we say that a space $X$ is a Michael space if X is a regular Lindelöf space such that $X\times ...

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61 views

### hyperspaces and selection principals

Two things bother me for which I haven't found an answer yet:
1.Is anyone familiar with an example of a topological space $X$, in which the hyperspace $2^X$ with the upper Fell topology is ...

**0**

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**1**answer

65 views

### subspace in pseudotopological space

Every topological space gives rise to a pseudotopological space. Conversely, if $X$ is a pseudotopological space then we can define a topology on $X$ such that every filter converging to $x$ in the ...

**5**

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**1**answer

373 views

### sets without perfect subset in a non-separable completely metrizable space

Suppose $X$ is a completely metrizable (but not separable) space. Suppose $D$ is a Borel (actually $F_{\sigma}$) subset of $X$. Is there any logical relation between the following statements?
[1] ...

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133 views

### Is the limit set of a group action always closed?

Let $G$ be a discrete group acting on a compact metric space $X$. A point $x\in X$ is called a limit point, if there is a base point $x_0\in X$ and an injective sequence $(x_k)_{k\in\mathbb{N}}$ in ...

**5**

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**1**answer

191 views

### Nonmetrizable compact totally disconnected spaces without isolated points

Brouwer proved that a topological space is homeomorphic to the Cantor set if and only if
1) it's non-empty,
2) it's compact,
3) it's totally disconnected,
4) it has no isolated points, and
5) ...

**3**

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**1**answer

208 views

### Continuous Functions

Is there a pair of continuous surjective functions say $f_1$ and $f_2$ from $\mathbb{Q}$ to itself such that for every $x, y \in \mathbb{Q}$, $f_1^{-1}(x) \cap f_2^{-1}(y)$ is non-empty?

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**3**answers

179 views

### Axiomatization of locally compact Hausdorff spaces via compact subspaces

The usual axiomatization of a topological space (in the sense of Bourbaki) goes by declaring certain subsets as being open and such that a few axioms are fulfilled by the family of open subsets.
It ...

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**2**answers

206 views

### An approximate infinite-dimensional fixed point theorem

Given $\epsilon > 0$ and $f : [0, 1]^{\omega} \rightarrow [0, 1]^{\omega}$, can we find $x$ such that $x \in \textrm{Conv}\left( \left\{f(y) : ||y - x||_{\infty} < \epsilon\right\}\right)$?
In ...

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**2**answers

165 views

### A set intersecting the graph of any continuous function in a finite set

New version of the problem I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true:
If $f :X\longrightarrow \Bbb{R}$ is continuous ...

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**0**answers

201 views

### Sequences and pseudocharacter in compact spaces

Is there a consistent example of a compact Hausdorff space $X$ on which the following holds?
i) there is a $Y \in {[X]}^{\aleph_1}$ such that $\psi (Y) = \aleph_1$; and
ii) there is no non-trivial ...

**5**

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**2**answers

424 views

### A question about the Stone–Čech compactification of discrete spaces

Let $D(\kappa)$ be the discrete space of cardinality $\kappa$, and $\beta D(\kappa)$ its Stone–Čech compactification.
Is there, for every infinite cardinal $\kappa$, a subset $Y \in [\beta ...

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**0**answers

265 views

### Linearly Lindelöf spaces with Lindelöf degree of uncountable cofinality

Is there a linearly Lindelöf space $X$ with
$\operatorname{cf} (L(X))> \aleph_{0}$
(where $L$ is the cardinal function Lindelöf degree)?
$L(X)$ must be a limit cardinal, like $\aleph_{\omega_{1}}$ ...

**8**

votes

**1**answer

326 views

### Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection).
A topological space $X$ is linearly ...

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**1**answer

129 views

### A space with countable tightness which is not a Fréchet space?

I need a space with countable tightness which is not a Fréchet space. In this space, I am searching for a point with no deleted neighborhood consisting entirely of P-points.
(A P-point is a point $x ...

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460 views

### Can closed compacts in a topological group behave “paradoxically” with respect to unions, intersections, and one-sided translations?

Consider two closed compacts $A$ and $B$ in a topological group $\Gamma$. Let $A'$ be a left translation of $A$ and $B'$ a left translation of $B$:
$A' = aA$,
$B' = bB$.
Suppose it is known that ...

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**1**answer

708 views

### Analysis of the boundary of the Mandelbrot set

Motivation: The Mandlebrot set is a simply connected set with an infinitely complex boundary, but CAN one move from interior to the exterior of this topological space by just crossing over a finite ...

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**1**answer

360 views

### Property of Mrowka

A topological space $X$ satisfies Property $K_1$ (Property of Mrowka) if the closure of the union of arbitrarily many $G_\delta$ sets of $X$ coincides with its sequential closure (the sequential ...

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**1**answer

243 views

### On the cardinality of perfect spaces with the countable chain condition

QUESTION: Does every regular perfect space with the countable chain condition have cardinality bounded above by the continuum? Is this at least true for perfectly normal ccc spaces?
Recall that a ...

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**2**answers

225 views

### Finitely cocomplete categories of compact Hausdorff spaces

Edit: Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the first part of question 1 ...

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**3**answers

344 views

### Existence of a Sub-Category of the Category of Topological Spaces

My question start with the following observations:
If you have a finite number of topological spaces $X_1, \dots , X_n$ you can define a space that is the disjoint union of its $\sqcup_{i=1}^n ...

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369 views

### Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?

Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...

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130 views

### Reference needed: Does pseudo laminated compact subsets of the plane separate the plane?

Hi,
doing my research I found the following problem and I´ll be glad if someone could give a reference.
We say that a compact connected subset $K$ of the plane is psuedo laminated if the following ...

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**2**answers

1k views

### Is $C^{\infty}[0,1]$ or $S$ separable?

I want to know if $C^{\infty}[0,1]$ or $S$ (Schwartz function space) is separable. Can somebody offer me some results or references?
Thank you!

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**6**answers

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### Topological characterization of the closed interval $[0,1]$

This question is related to question 92206 "What properties make $[0, 1]$ a good candidate for defining fundamental groups?" but is not exactly equivalent in my opinion. It is even suggested in one ...

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**2**answers

245 views

### What are Normal Sets (Fréchet)?

In 1913, LEJ Brouwer started a new approach to give a topologist's definition of the notion dimension ("Über den natürlichen Dimensionsbegriff", Journal für die reine und angewandte Mathematik, 142, ...

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**4**answers

891 views

### Compactness of the Hilbert cube without the Axiom of Choice

I am just curious: is there a published proof of the compactness of the Hilbert cube that does not use the Axiom of Choice, or is it well known?

**1**

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**1**answer

498 views

### Example of a topological space

In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such ...

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**1**answer

84 views

### Question regarding closure of sets defined by the vanishing of holomorphic functions

Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, ...

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**1**answer

216 views

### Relative extremely disconnected space

A topological space $X$ is called relative extremely disconnected if it has a base $B$ (for open subsets) such that disjoint elements in $B$ have disjoint closure.
Does it exist an infinite ...

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**3**answers

754 views

### Zero-dimensional space

Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that ...

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**1**answer

347 views

### Extremely disconnected space

A topological space $X$ is called relative extremely disconnected if it has a base $B$(for open subsets) such that disjoint elements in B have disjoint closure, i.e, if $C, D$ in $B$ and $C\cap ...

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**1**answer

337 views

### Does there exist a countable partition of [0,1] by disjoint closed subsets? [duplicate]

Possible Duplicate:
Why are the integers with the cofinite topology not path-connected?
As in the title, is it possible to find closed, disjoint subsets $C_n$ of $[0,1]$ such that $[0,1] = ...

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**1**answer

244 views

### Ultralimit versus partial limit

Let $\omega$ be a nonprincipal ultrafilter on $\mathbb N$.
A standard construction gives an $\omega$-limit, say $x_\omega$, for any bounded sequence $(x_n)$ of real numbers.
Namely, there is unique ...

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**2**answers

314 views

### Fundamental groups and homology groups of closed subsets of the plane

Let $X$ be a closed subset of $\mathbb{R}^2$. What restrictions are there on $\pi_1(X)$ and on the homology groups of $X$ (both singular and Cech)? This is elementary if $X$ has reasonable local ...

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**1**answer

420 views

### Constructible sets in Hausdorff spaces

In an attempt to write a proof by contradiction, I end up with a space $X$ with the following properties:
(0) $X$ is nonempty,
(1) $X$ is Hausdorff,
(2) $X$ has no isolated points,
(3) every subspace ...

**5**

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**2**answers

633 views

### Cantor Sets Inside Cantor Sets

(Or: "I heard you liked Cantor Sets...")
I'm working on a student project, and the following construction came up very naturally: If $C$ is the usual Cantor Set, build a countable union of copies of ...

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**3**answers

557 views

### Topological spaces determined by generalized metric spaces

At http://en.wikibooks.org/wiki/Real_Analysis/Metric_Spaces you can find the standard definition of a metric space: a set $X$ given with a function $d:X\times X\to\mathbb{R}$ that satisfies properties ...

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**2**answers

252 views

### constructing a curve dividing two sets of points

Lets assume I have two sets of points, each characterized as being "A" or "B", respectively, that are in a Euclidean plane. Theoretically these two sets are samplings from a space that has ...

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**4**answers

480 views

### Existence of an arbitrary Small positive continuous real Valued Function

Let $(X,\tau)$ be a Tychonoff Topological space.
For each $x\in X$ consider an arbitrary positive real number $\epsilon_x>0$. Is There a continuous real valued function $f:X\rightarrow ...

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**1**answer

343 views

### On One point Lindeloffication of topological spaces

As you Know when we define a topological space to be the one point compactification of the topological space $X$, we look for a compact space $Y$ such that $X\subset Y$ and $X$ is dense in $Y$ and ...

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517 views

### On Pseudo-finite topological spaces

We recall that a topological space $(X,\tau)$ is Pseudo-finite, if each compact subset of $X$ is finite.
One of the classical example of Pseudo-finite topological spaces can be considered as an ...

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**1**answer

263 views

### Compact subsets and Hausdorffness of Topology

We Know that all closed subsets of a compact topological space $(X,\tau)$ are compact. But if we add The Hausdorff condition on the topology $\tau$ we could see the equivalence of these subsets.(i.e. ...