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7 votes
0 answers
349 views

An open set which is not the union of a closed set and a countable set

The following fact is probably a known result: Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set. Proof:...
9 votes
1 answer
410 views

On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces

In the known paper On the reconstruction of topological spaces from their group of homeomorphisms by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are ...
6 votes
0 answers
255 views

Every Polish space is the image of the Baire space by a continuous and closed map, reference

The following result was originally proven by Engelking in his 1969 paper On closed images of the space of irrationals (AMS, JSTOR, MR239571, Zbl 0177.25501) Every Polish space (i.e. every separable ...
4 votes
0 answers
142 views

Consistency of a strange (choice-wise) set of reals, pt. 2

This is a follow-up on this question. Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology Every countable family of non-empty pairwise disjoint subsets of $...
7 votes
2 answers
722 views

Consistency of a strange (choice-wise) set of reals

Consider a set $X\subseteq \mathbb{R}$ such that $X$ is not separable wrt its subspace topology For all $r\in\mathbb{R}$ there exists a sequence $(x_n)_{n\in\omega} \subset X$ converging to $r$ In a ...
4 votes
1 answer
279 views

Product topology from two premetric spaces induced by sum of premetrics?

For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$. Do ...
4 votes
1 answer
195 views

Consistency of the Hurewicz dichotomy property

Just to fix the environment, let's work in the Baire space $\omega^\omega$, the space of infinite sequences of natural numbers with the product of the discrete topology over $\omega$. We say that a ...
3 votes
1 answer
192 views

Co-analytic $Q$-sets

A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
4 votes
1 answer
259 views

Reference request: large generalized probability measures

I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size? I'...
6 votes
1 answer
205 views

Topological complexity of ultrafilters in $2^\kappa$ for uncountable $\kappa$

It is a well known fact that if $\mathcal{F}$ is a non-principal ultrafilter on $\omega$, then the set $\{ \alpha \in 2^\omega : \alpha \in \mathcal{F}\}$ (conflating binary strings with subsets of $\...
8 votes
3 answers
937 views

BCT equivalent to DC

Do you know where I can find proof of equivalence Baire Category Theorem and DC (Axiom of Dependent Choice)? It is well known fact but I can't find appropriate literature with the proof.
2 votes
0 answers
120 views

Two small uncountable cardinals related to Q-sets

A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$. Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
1 vote
1 answer
1k views

A new generalisation of dimension? part 2

I worked this theory : A new generalization of the dimension? I have a theorem about dimensions which is more general and simple than for matroids. Definition 1: A structure $S$, is a pair $(X, \...
4 votes
0 answers
105 views

Borel selections of usco maps on metrizable compacta

The problem posed below is motivated by this problem of Chris Heunen and in fact is its reformulation in the language of usco maps. Let us recal that an usco map is an upper semicontinuous compact-...
3 votes
1 answer
436 views

Stone topological Boolean algebras

I am looking for an initial reference for a theorem which is known, namely: Theorem: A Boolean algebra $A$ admits a Stone space topology (i.e. is the underlying algebra of a Stone topological ...
12 votes
1 answer
316 views

A reference to a theorem on the equivalence of ideals of measure zero in the Cantor cube

I am looking for a reference of the following (true) fact: Theorem. For any two continuous strictly positive Borel probability measures $\mu,\lambda$ on the Cantor cube $2^\omega$ there exists a ...
17 votes
1 answer
988 views

Can two-point sets be Borel?

Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914. I wonder if the following question of ...
10 votes
2 answers
750 views

Is there a compact space with no countably generated dense subspace?

This is a reformulation of this MO question which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give ...
3 votes
1 answer
186 views

Cardinality of connected subspaces

Is there a cardinal $\kappa>2^\omega$ and a connected space $X$ such that (1) $|X|=\kappa$, and (2) every connected subset of $X$ (with at least 2 points) has cardinal $\kappa$? Let's assume ...
7 votes
1 answer
389 views

References for higher descriptive set theory surveys

A student of Adi Jarden and mine attempts at generalizing results on selection principles from the Baire space $\omega^\omega$ to the higher Baire space $\kappa^\kappa$ ($\kappa$ uncountable), and ...
7 votes
0 answers
266 views

Remote points in $\beta X$

It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space $...
4 votes
5 answers
1k views

A generalized diagonal?

A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It ...
36 votes
4 answers
4k views

How far is Lindelöf from compactness?

A while ago I heard of a nice characterization of compactness but I have never seen a written source of it, so I'm starting to doubt it. I'm looking for a reference, or counterexample, for the ...
23 votes
3 answers
2k views

An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request

There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter. But there's ...
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 ...
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 ...