# Tagged Questions

**2**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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] ...

**5**

votes

**1**answer

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

**1**answer

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

**3**answers

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 ...

**3**

votes

**2**answers

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

**0**answers

170 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**

votes

**2**answers

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

**0**answers

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

**1**answer

308 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

**1**answer

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**

votes

**0**answers

442 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 ...

**0**

votes

**1**answer

355 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

**1**answer

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

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**12**

votes

**5**answers

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

**1**answer

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 ...

**2**

votes

**1**answer

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

**1**answer

319 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 ...

**9**

votes

**1**answer

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 ...

**7**

votes

**3**answers

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 ...

**5**

votes

**4**answers

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

**1**answer

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

**4**answers

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

**1**answer

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

**1**answer

244 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

**1**answer

225 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

**1**answer

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

**2**answers

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

**1**answer

317 views

### Fuzzy topology : references [closed]

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

**5**

votes

**0**answers

355 views

### Continuous images of Cantor cubes

The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more ...

**5**

votes

**1**answer

352 views

### Products with compactly generated spaces

It is well known that if $X$ and $Y$ are topological spaces with $X$ locally compact Hausdorff and $Y$ compactly generated, then $X \times Y$ (with the ordinary product topology) is compactly ...

**1**

vote

**0**answers

128 views

### Local cartesian closedness in the category of compactly generated spaces

According the the nLab, the category of compactly generated (CG) spaces is not locally cartesian closed.
So if $A$ is a CG space and $C$ a CG space above $A$, $C$ may not be exponentiable.
What if we ...

**7**

votes

**2**answers

632 views

### Patching together homeomorphisms: how badly can it fail?

Suppose we have a set $X$ with $X=U \cup V$. If we pick a permutation $f$ of $U$ and a permutation $g$ of $V$ which agree on the intersection $U \cap V$, we can coalesce them into one big endo-map $F$ ...

**3**

votes

**0**answers

339 views

### Lifting local compactness to a covering space

(I decided to repost this from MathSE, since the question seems to not be as easy as I had thought)
NB: In this question, local compactness is used in its weak form, i.e. in a locally compact space, ...

**10**

votes

**0**answers

917 views

### Paracompact Hausdorff but not compactly generated?

I'm sorry to be asking a (possibly) elementary question, but I've run into a problem in point-set topology; I've just read that there exists paracompact Hausdoff spaces which are not compactly ...

**3**

votes

**1**answer

924 views

### $\Delta_{2}^{1}$-hard set?

Hello everybody!
I'm interested in $\Delta_{2}^{1}$ subsets of Polish spaces, i.e. those sets that are both $\Pi_{2}^{1}$ and $\Sigma_{2}^{1}$ in the boldface hierarchy of Polish spaces.
There is a ...

**12**

votes

**4**answers

2k views

### Locally compact Hausdorff space that is not normal

What is a good example of a locally compact Hausdorff space that is not normal? It seems to be well-known that not all locally compact Hausdorff spaces are normal (and only a weaker version of ...

**25**

votes

**4**answers

903 views

### Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?

In his 1967 paper A convenient category of topological spaces,
Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces
as a good replacement of the catgory Top topological ...

**2**

votes

**0**answers

60 views

### Characterizing local homeomorphisms into an exponent

Let $X$,$Y$, and $Z$ be (compactly generated) spaces. Suppose $f:Z \to Y^X$ is a local homeomorphism. How can we tell this from its adjoint $\tilde f:Z \times X \to Y$? I.e., I want a property $P$ ...

**9**

votes

**10**answers

2k views

### Are nets and filters useful in geometry and topology?

Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...

**0**

votes

**4**answers

2k views

### Does Cauchy continuity imply uniform continuity? [No.] [on hold]

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X ...

**4**

votes

**0**answers

435 views

### continuous selection of a multivalued function?

The title is probably a bit too broad. I frequently encountered the following situation: suppose I need to select a solution to a linear equation from a compact set. Can I make this selection ...

**3**

votes

**2**answers

873 views

### Paracompact but not Hausdorff

Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.

**38**

votes

**4**answers

2k views

### How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?

This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...