This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable partition of the unit interval $[0,1]$ into closed subsets exactly one set of the partition is non-empty...

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$? Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals: $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite inter...

The Baire category theorem implies that a nonempty complete metric space without isolated points must be uncountable. In many situations I have encountered, the "natural examples" of complete metric spaces without isolated points (of a certain type, or possibly with some additional structure) in...

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal U$ of $X$ has a disjoint refinement $\mathcal V\subset\mathcal B$. It can be shown that (1...

This question assumes familiarity with combinatorial cardinal characteristics of the continuum. Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing enumeration. Thus, for each natural number $n$, $a(n)$ is the $n$-th element of $a$ in increasing order. This way, every infinite se...

Beyond measure zero we have yet another measure-y notion of smallness: strong measure zero. A set $S\subseteq\mathbb{R}$ is strong measure zero if, for any $f:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, there is a sequence $U_i$ of open sets with the diameter of $U_i$ is $<f(i)$, and $S\subseteq\big...

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-set in $\mathbb R$. It can be shown that $q_0$ is the smallest cardinality of a set $A\subset \...

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ is continuous for every $C\in\mathcal C$. Theorem 7.0 and Lemmas 7.4, 7.5 from Todorcevic's boo...

A tower is a subset $T\subset [\omega]^\omega$ of the family $[\omega]^\omega$ of all infinite subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has no infinite pseudointersections. A tower is regular if its cardinality is a regular cardinal. C...

Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-closed set $A \subset X$ and for every point $x \in \overline{A} \setminus A$ there is a sequenc...

A non-empty topological space without isolated points is called maximal if every finer topology on that space has at least an isolated point. The existence of a (Hausdorff) maximal space is a simple consequence of Zorn's Lemma. Note that in a maximal space $(X, \tau)$, nowhere dense sets are cl...

I was wondering whether it is consistent to have $\frak{c} = \aleph_{\frak{c}}$ where $\frak{c} = 2^{\aleph_0}$ is the cardinality of the reals (over ZFC). If so, what interesting consequences of this statement are known (besides ¬CH)? I was curious about this because in some sense $\frak{c}$ is ...

In "The Kunen-Miller Chart (Lebesgue Measure, The Baire Property, Laver Reals and Preservation Theorems for Forcing)" by Haim Judah and Saharon Shelah JSL Vol. 55, No. 3 (Sep., 1990), pp. 909-927 ([JdSh308] in Shelah's numbering) the authors remark on the final page that adding a pair of Hechler ...

We can define a very simple cardinal characteristic in the following way. Recall the relation $\leq^*$ on $\omega^\omega$ defined by $x\leq^* y$ iff $x(i)\leq y(i)$ for all but finitely many $i$. For $x,y\in\omega^\omega$, say that $x$ and $y$ are comparable, denoted by $x\parallel y$, if either ...

There are a few jokes in "Do good math jokes exist?" that I think are inappropriate. Specifically: Nov 21 '09 at 9:01 item (1) Oct 19 '09 at 16:03 Dec 10 '09 at 23:14 Dec 15 '09 at 1:31 All of these are jokes about people's bodies (mostly women's bodies) with sexual implications. In short, t...