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0
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1answer
72 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 ...
2
votes
0answers
45 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 ...
2
votes
1answer
107 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 ...
0
votes
0answers
51 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
votes
0answers
48 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 ...
4
votes
1answer
310 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] ...
4
votes
0answers
114 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
votes
1answer
167 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
votes
1answer
201 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?
1
vote
3answers
161 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 ...
4
votes
2answers
192 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 ...
3
votes
2answers
156 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 ...
2
votes
0answers
167 views
+50

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
votes
2answers
352 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 ...
9
votes
0answers
240 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
1answer
307 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 ...
1
vote
1answer
119 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 ...
16
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0answers
441 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 ...
10
votes
1answer
650 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 ...
0
votes
1answer
354 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 ...
6
votes
1answer
220 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 ...
2
votes
2answers
211 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 ...
0
votes
3answers
340 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 ...
19
votes
0answers
350 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 ...
1
vote
0answers
127 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 ...
3
votes
2answers
931 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!
12
votes
5answers
1k views

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 ...
1
vote
2answers
233 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, ...
12
votes
4answers
861 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
vote
1answer
481 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 ...
0
votes
1answer
83 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, ...
2
votes
1answer
209 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 ...
0
votes
1answer
318 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 ...
7
votes
1answer
224 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 ...
3
votes
2answers
304 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 ...
9
votes
1answer
407 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
votes
2answers
592 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 ...
7
votes
3answers
529 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 ...
1
vote
2answers
235 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 ...
5
votes
4answers
473 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 ...
3
votes
1answer
334 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 ...
6
votes
4answers
503 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 ...
1
vote
1answer
255 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. ...
0
votes
1answer
243 views

Lindelöf subsets of $P$-spaces

A completely regular topological space $(X,\tau)$ is called a $P$-space, if every $G_\delta$-subset of $X$ is open .$\($i.e $\tau$ is closed under countable intersection$\)$. Here we recall some ...
2
votes
1answer
224 views

Existence of a non-submetrizable topological space $(X, \tau)$

We recall that a topological space $(X,\tau)$ is submetrizable, if there is a coarser metrizable topology $\tau'$ that $\tau\supseteq\tau'$. one of the properties of these topological spaces is ...
1
vote
1answer
144 views

Counterexample about Jones lemma with special weak condition.

Jones Lemma is One scale about recognizing that a topological space is not normal. This lemma tells us, if The topological space $X$ has a dense subset $D$ and a closed discrete subset $S$, with the ...
5
votes
2answers
309 views

Existence of a compactification of $\mathbb{R}$ with $\aleph_0$ remainder

We now that the space $\mathbb{R}$ has compactifications with one point reminder, and two point reminder. but there is no compactification of $\mathbb{R}$ with three point reminder and the same holds ...
2
votes
1answer
193 views

Partitions of an interval

This question asks about properties of functions which are "piecewise" polynomials. I would like to ask a specific question about the meaning of "piecewise" there. Specifically, consider "partitions" ...
21
votes
4answers
1k views

is f a polynomial provided that it is “partially” smooth?

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$. Suppose for each $n\in ...
2
votes
1answer
386 views

Filling $\mathbb{R}^3$ with skew lines

I would like to know if it is possible to fill $\mathbb{R}^3$ with lines with the following two properties: (1) Every point $x \in \mathbb{R}^3$ is contained in precisely one line. (2) Every ...