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6
votes
4answers
521 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 ...
2
votes
1answer
264 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
264 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
234 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
166 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
316 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
205 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" ...
23
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
430 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 ...
-1
votes
1answer
342 views

Fuzzy topology : references [closed]

Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
5
votes
0answers
367 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
1answer
367 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 ...
3
votes
0answers
235 views

Coloring $\mathbb{Z}^k$ and a fixed point theorem

This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...
1
vote
0answers
132 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
2answers
636 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 $F$ ...
10
votes
1answer
370 views

convexity of images of space-filling curves

Suppose $f:[0,1]\to[0,1]^2$ is continuous and for each $t\in[0,1]$, the area of $\lbrace f(s) : 0\le s\le t \rbrace$ is $t$. For what sets of values of $t\in[0,1]$ can $\lbrace f(s) : 0\le s\le t ...
1
vote
1answer
361 views

Is a section of a proper map proper?

Suppose $f\colon X \rightarrow Y$ is a continuous map of topological spaces and $s\colon Y \rightarrow X$ is a continuous section to $f$, i.e., $f\circ s = 1$. If $f$ is proper does this mean that $s$ ...
1
vote
0answers
122 views

Follow up question on the measure of the difference between a partial selector and a selector…

This is a different question from my previous question Difference between a partial selector and a selector, however I am going to repeat the preamble... In Kharazishvili's "Nonmeasurable Sets and ...
0
votes
1answer
137 views

Difference between a partial selector and a selector…

In Kharazishvili's "Nonmeasurable Sets and Functions" there is the following theorem: There exists a subset $X$ of $\mathbb{R}$ which is a Vitali set and a Bernstein set. The proof is as ...
1
vote
2answers
305 views

Cardinality of the set of countable dense subgroups of the reals up to isomorphism.

Joel David Hamkins in an answer to my question Countable Dense Sub-Groups of the Reals points out that "one can find an uncountable chain of countable dense additive subgroups of $\mathbb{R}$ whose ...
3
votes
0answers
354 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, ...
11
votes
0answers
938 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
1answer
928 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 ...
0
votes
1answer
242 views

Random question: Is there a set-theoretic description of projective space? [closed]

I met projective space via a recent class on perspective drawing, believe it or not, but I didn't know that this was the "space" we were using. I came across a more detailed description trawling the ...
13
votes
5answers
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 ...
10
votes
2answers
1k views

Does Riemann map depend continuously on the domain?

I've always taken this for granted until recently: In the simplest case, given Jordan curve $C \subseteq \mathbb{C}$ containing a neighborhood of $\bar{0}$ in its interior. Given parametrizations ...
19
votes
0answers
938 views

Subfields of $\mathbb{C}$ isomorphic to $\mathbb{R}$ that have Baire property

While sitting through my complex analysis class, beginning with a very low level introduction, the teacher mentioned the obvious subfield of $\mathbb{C}$ isomorphic to $\mathbb{R}$, and I then ...
27
votes
4answers
954 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
0answers
61 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 $P$ ...
13
votes
2answers
870 views

Complete metric on the space of Jordan curves?

I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and ...
15
votes
0answers
1k views

Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...
5
votes
4answers
3k views

Is a topology determined by its convergent sequences?

Just a basic point-set topology question: clearly we can detect differences in topologies using convergent sequences, but is there an example of two distinct topologies on the same set which have the ...
9
votes
10answers
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 ...
1
vote
3answers
1k views

Definition of Connected Subspace

In Munkres (Chapter 3, Section 23, p. 148), Munkres shows that if a subspace $Y$ of a space $X$ is not connected, then there are two disjoint open subsets $A,B$ such that the union of $A$ and $B$ ...
8
votes
1answer
1k views

The Wedge Sum of path connected topological spaces

A definition of wedge sum can be found here: http://en.wikipedia.org/wiki/Wedge_sum My professor has claimed that wedge sums of path connected spaces X and Y are well-defined up to homotopy ...
0
votes
4answers
2k views

Does Cauchy continuity imply uniform continuity? [No.] [closed]

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
0answers
445 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
2answers
901 views

Paracompact but not Hausdorff

Do paracompact non-Hausdorff spaces admit partions of unity? I'm just curious.
60
votes
9answers
15k views

solving f(f(x))=g(x)

This question is of course inspired by the question How to solve f(f(x))=cosx and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a ...
4
votes
4answers
927 views

What is a reference for profinite sets?

The question is in the title. The motivation behind the question is as follows: there are plenty of references about profinite groups and profinite completions of groups. It seems that their not ...
45
votes
4answers
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. ...