12
votes
1answer
224 views

Discrete subsets in the topology of pointwise convergence vs. metrisability

While reading Arkhangel'skii's Topological function spaces, I encountered an unexpected application of Martin's Axiom. This is Theorem II.5.20: Assume $\mathsf{MA}+\neg \mathsf{CH}$. Let $X$ be a ...
5
votes
2answers
328 views

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold? The answer is negative, and in the interests of self-contained ...
8
votes
2answers
319 views

Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$. As a C$^*$-algebraist, I ...
5
votes
2answers
280 views

non-Borel set which intersects every compact in a Borel set

I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...
5
votes
1answer
139 views

A realcompact analogue of the Baire category theorem

Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...
8
votes
1answer
279 views

Is $\ell^\infty$ Polishable?

Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
3
votes
1answer
97 views

Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model ...
2
votes
0answers
317 views

Topological proof of a result in Logic

I proved the result below using logic. My questions: Can this theorem be proved by purely topological means? Do you know any theorems that either can be used to prove the same result, or which give ...
7
votes
1answer
367 views

Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...
3
votes
1answer
269 views

Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.
6
votes
1answer
401 views

Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that are (a) ...
16
votes
2answers
470 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
3
votes
0answers
181 views

Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
6
votes
1answer
388 views

Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?

Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition: The meager sets are sets which are ...
16
votes
2answers
732 views

An order type $\tau$ equal to its power $\tau^n, n>2$

(This is a re-post of my old unanswered question from Math.SE) For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types. Recall that: ...
2
votes
1answer
133 views

Is there a maximal (or maximal Tychonoff) non normal space?

Is there a maximal (or maximal Tychonoff) non normal space? In "A Problem of Set-Teoretic Topology" the existence of a maximal Tychonoff space is asserted. Also there exists a perfectly normal maximal ...
16
votes
2answers
505 views

Intersection of compact sets in the unit interval

Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr ...
4
votes
2answers
295 views

Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$

Let $X$ be the real line with the usual topology. Then clearly $|C(X)| = c = |X|$ and on the other hand $|X| = 2^{\aleph_0}$. Now my question is as in the title: Is there a Tychonoff space $X$ of ...
7
votes
1answer
555 views

What is the shape of mathematical universe?

Shape? At the usual mathematical literature when we can discuss about the shape of a "space" that we have a kind of "topography" on it. For example a topology, metric, geometry, etc. Note that for ...
2
votes
2answers
192 views

When is a filter generated by a (countable) chain?

In any partial order $(P,\leq)$ it is easy to see that every chain generates (i.e., by taking the upwards closure) a filter, and any filter built as a result of the Rasiowa-Sikorski lemma in forcing ...
10
votes
1answer
207 views

Idempotent ultrafilters and the Rudin-Keisler ordering

Short version: what can we say about the place of idempotent ultrafilters in the Rudin-Keisler ordering? Longer version: If $U$, $V$ are (nonprincipal) ultrafilters on $\omega$, then we write ...
16
votes
1answer
462 views

Question about product topology

Suppose $S\subset\mathbb{R}$ is dense without interior point, and for every open interval $I,J\subset\mathbb{R}$, $I\cap S$ is homeomorphic to $J\cap S$. Is $S\times S$ homeomorphic to $S$? By Luzin ...
11
votes
2answers
416 views

How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?

How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$? How many subsets of the long line $\omega_1\times[0,1)$ are order isomorphic to $\mathbb{Q}$? I can see that results in ...
10
votes
1answer
410 views

Restrictions of null/meager ideal

Let I denote the null (resp. meager) ideal on reals. Is it consistent that for any pair of non null (resp. meager) sets A and B, there is a null (resp. meager) preserving bijection between A and B? In ...
5
votes
1answer
446 views

Showing a filter with a certain property on the power set of $\mathbb{Z}$ is a one point filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
4
votes
0answers
164 views

Well-founded families of sets and topological convergence

Background/Motivation A space is scattered if every non-empty subset has an isolated point. A space is pseudoradial if every non-closed set contains a transfinite sequence (a well-ordered net) ...
6
votes
4answers
428 views

Periodic point-free maps and free ultrafilters.

Let $X$ be a set and $u$ be a free ultrafilter on $X$. We can consider a topology on $X$ by declaring every element of $u \cup \{\emptyset \}$ to be open. El'kin's original motivation for looking at ...
7
votes
1answer
377 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 ...
8
votes
3answers
782 views

Axiom of Choice and continuous functions

Do you know if the following statement is an equivalent form of the axiom of choice or not? If $X$ is a compact metric space, then every continuous function $f: X \longrightarrow \mathbb{R}$ is ...
3
votes
1answer
166 views

A question on hereditary Lindelof number

Let $X$ be space. A space $X$ is called right-separated if it can be well-ordered in such a way that every initial segment is open in $X$. See the related link (left-separated). How could we show ...
5
votes
1answer
205 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
votes
0answers
253 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
1answer
144 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 ...
1
vote
0answers
221 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 $X_\alpha$ ...
5
votes
3answers
228 views

Well-ordering with a topological property

Assuming the axiom of choice, is there a well-ordering of the reals such that every initial segment is closed for the usual topology? If the continuum hypothesis helps, we can also assume it. An ...
2
votes
1answer
405 views

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 $\mu(X)=1$. ...
5
votes
1answer
303 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 ...
8
votes
0answers
470 views

Is every $\sigma$-algebra of sets *abstractly* the Borel algebra of a topology on perhps some other set?

Is every sigma-algebra the Borel algebra of a topology? inspires the present question which asks for less. Question: Given a $\sigma$-algebra ${\cal A}$ on a set $X$, does there exist a topology ...
5
votes
0answers
315 views

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 ...
5
votes
1answer
291 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 ...
11
votes
1answer
719 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 ...
8
votes
0answers
504 views

In ZF, when is a disjoint union of metrizable spaces metrizable?

It is easy to see that the disjoint union $\bigsqcup_i X_i$ of a collection of metric spaces is metrizable, simply by rescaling or chopping off the individual metrics to have diameter at most one, and ...
0
votes
1answer
252 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 ...
8
votes
0answers
537 views

Does Urysohn's Lemma imply Dependent Choice?

It's widely known$^{1}$ that in the proof of Urysohn's Lemma (UL) one uses the Principle of Dependent Choice (DC). Inspired by the equivalence between DC and Baire's Category Theorem$^{2}$, I'd like ...
5
votes
2answers
215 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?
44
votes
2answers
3k views

Is every sigma-algebra the Borel algebra of a topology?

This question arises from the excellent question posed on math.SE by Salvo Tringali, namely, Correspondence between Borel algebras and topology. Since the question was not answered there after some ...
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 ...
2
votes
1answer
192 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 ...
2
votes
1answer
255 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 ...
3
votes
1answer
350 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 < = $2^{\aleph_0}$?