All Questions
Tagged with set-theory gn.general-topology
433 questions
4
votes
1
answer
255
views
Forcing over the poset of nonempty open subsets of a nice topological space
Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, ...
- 2,550
1
vote
0
answers
321
views
Type I subspaces of the Stone Cech compactification of $\omega$
EDIT: I found a construction, see below. I decided not to delete the question in case someone is interested.
A space $X$ is of Type I if $X=\cup_{\alpha<\omega_1} X_\alpha$, where each $X_\alpha$ ...
- 1,855
2
votes
0
answers
371
views
Descriptive set theory on $\mathbb{R}^\mathbb{N}$
The short version of my question is, What is a good source for learning about descriptive set theory on the space $\mathbb{R}^\mathbb{N}$, under the product topology coming from the discrete topology ...
- 20.9k
3
votes
1
answer
164
views
Algebras with countable chains only
Is there an example of an uncountable Boolean algebra $B$ in which every chain is countable and such that $\ell_\infty$ embeds into the Banach space $C(\mbox{Stone }B)$? The latter requirement is not ...
- 3,937
4
votes
1
answer
668
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special extremally disconnected spaces with only finite isolated points
We Know that a cardinal $\kappa$ is measurable if there is a set $X$ with cardinal $\kappa$ and a {0,1}-measure $\mu: P(X) \rightarrow ${$0,1$} so that for all $x \in X$, $\mu(x)=0$ and $\mu(X)=1$. ...
- 14k
6
votes
1
answer
375
views
How much $\beta \mathbb{N}$ is homogenous?
Let $p,q\in \beta \mathbb{N}\setminus \mathbb{N}$. Must always the spaces $\beta \mathbb{N}\setminus \{p\}$ and $\beta \mathbb{N}\setminus \{q\}$ be homeomorphic? If no, can we for each point $p\in \...
- 11.5k
7
votes
0
answers
466
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Closure properties of familes of $G_\delta$ sets.
Given a family of sets $G\subset P(X)$, can one characterize by "closure properties" alone whether or not $G$ arises as the family of all $G_\delta$ for some topology on $X$? some Polish space ...
- 17.6k
6
votes
1
answer
634
views
Arbitrary small positive lower semi continuous functions
This question is a generalization of the question posed in this page to lower semi continuous functions. so let me describe the Question in the following way.
Def: Let $(X,\tau)$ be a Tychonoff ...
CommunityBot
- 1
16
votes
1
answer
2k
views
Characterization of Stone-Cech compactifications
Suppose I have an infinite discrete topological space $X$ of cardinality $\kappa$. Then I know some things about the Stone-Cech compactification, $\beta X$: it is Hausdorff and compact but not ...
- 42.4k
0
votes
1
answer
278
views
On the compactness of a certain chain topology [closed]
Let $X$ be a non-empty set and $I$ a collection of some nested subsets of $X$ indexed by a linearly ordered set $(\Lambda,\le)$ such that $I$ always contains the void set $\emptyset$ and the whole set ...
- 236k
6
votes
2
answers
257
views
Borel functions on $\omega_1$
Endow $\omega_1$ with order topology. It is easy to show that each continuous function $f\colon \omega_1\to \mathbb{R}$ is eventually constant. Is the same true for Borel functions?
- 73.2k
2
votes
1
answer
220
views
Extending BAs to weakly countably distributive algebras.
Suppose $\mathscr{A}$ is a complete Boolean algebra (assume it is c.c.c. if you wish). Is there any, say, canonical embedding $\mathscr{A}\subseteq \mathscr{B}$ into a complete Boolean algebra which ...
- 236k
2
votes
1
answer
274
views
Does X have any diagonal properties?
Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the support of $y$, and let ...
- 11.5k
4
votes
1
answer
399
views
If a topological space X has $\aleph_1$-calibre, then it must be star countable?
If a topological space X has $\aleph_1$-calibre[definition], then it must be star countable?
What if the cardinality of the topological space X is additionally < = $2^{\aleph_0}$?
CommunityBot
- 1
0
votes
1
answer
501
views
7
votes
2
answers
597
views
A characterisation of well-ordering ?
It is easy to prove that if $E$ is well-ordered, and if $f$ is a strictly increasing map from $E$ to $E$, then, for all $x$ in $E$, $f(x) \ge x$ (just consider the sequence $x$, $f(x)$, $f(f(x))\dots$)...
- 39.9k
1
vote
0
answers
150
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 ...
CommunityBot
- 1
0
votes
1
answer
194
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 follows:
...
- 73.2k
1
vote
2
answers
405
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 ...
CommunityBot
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2
votes
1
answer
404
views
Follow up question on union of disjoint Vitali sets...
Since I haven't received a satisfactory answer to my initial question I'm going to ask a somewhat weaker one...
This time we say $X$ is a Vitali set in the closed interval $[0, 1]$ with respect to $\...
- 54.2k
7
votes
1
answer
2k
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Universally measurable sets and weak topology
After I posted this question, a couple of months ago, and got from MO-users several
good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...
CommunityBot
- 1
6
votes
2
answers
492
views
Distinct, non-homeomorphic, profinite topologies on a given abstract group ?
Just a silly little question which arose in connection with infinite Galois groups and their Krull topology:- can a given abstract group be endowed with distinct, non-homeomorphic, profinite ...
- 1,012
3
votes
1
answer
1k
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$\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 ...
- 236k
5
votes
1
answer
523
views
Injections to binary sequences that preserve order
Suppose we have a countable set S with a total order. Can we give an injection from S to the set of finite binary sequences that end in all zeros that preserves the ordering? The order on binary ...
- 85.6k
3
votes
0
answers
251
views
What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
11
votes
2
answers
721
views
Inconsistency and workaday independence.
Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
- 2,762
3
votes
1
answer
617
views
How to make an ultranet
The only examples of ultranets/ultrafilters described in Bourbaki and Willard are the trivial ones (generated by a single point). I know that their existence relies in general on the axiom of choice ...
- 44.4k
3
votes
1
answer
4k
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A sequence with no convergent subsequence without choice
By Tychonoff Theorem $\prod_{\mathbb R} [0,1]$ is compact and since $\mathbb R=2^{\omega}$, if for $\alpha \in 2^{\omega}$, $x_n(\alpha)=\alpha(n)$ then if we consider a subsequence $x_{n_0}, x_{n_1}, ...
- 16.2k
4
votes
2
answers
452
views
A family of subsets with a "gluing" property
Somewhat in line with this previous MathOverflow question:
I'm looking at a combinatorial structure consisting of a finite set $S$ of objects, and a family $F$ of designated subsets of $S$. We call ...
CommunityBot
- 1
3
votes
3
answers
444
views
Shape of long sequences in C(ω_1)
Apologies for the vague title - I couldn't come up with a single sentence that summarised this problem well. If you can, please edit or suggest a better one!
This question is also rather specific and ...
- 1,331
5
votes
0
answers
558
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 ...
- 1,839
-3
votes
2
answers
314
views
Dispensing with the notion of infinity for the sake of coverings [closed]
Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but ...
- 9,201
11
votes
5
answers
1k
views
Confusion over a point in basic category theory
"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological ...
- 65.4k