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2
votes
2answers
265 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
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 ...
20
votes
0answers
408 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
131 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
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!
14
votes
6answers
2k 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
270 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, ...
15
votes
5answers
1k 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
517 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
85 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
217 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 ...
1
vote
3answers
828 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 ...
0
votes
1answer
379 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 ...
1
vote
1answer
462 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] = ...
7
votes
1answer
281 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 ...
4
votes
2answers
343 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
445 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 ...
6
votes
2answers
713 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 ...
8
votes
3answers
636 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
275 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
485 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
375 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
536 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
282 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
273 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
257 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 ...
2
votes
1answer
193 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
336 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
220 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
459 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
366 views

Fuzzy topology : references [closed]

Hey. I'm looking for references in fuzzy topology. Does anyone know a good book ?
5
votes
0answers
425 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
393 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
239 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 ...
2
votes
0answers
137 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
644 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
372 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
367 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
124 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
308 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
367 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
962 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
929 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
248 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 ...
14
votes
5answers
3k 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 ...
20
votes
0answers
968 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 ...
31
votes
4answers
1k 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 ...