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A subset $X\subset \mathbb R$ is called $\bullet$ a $Q$-set if for every subset $A\subset X$ there exists a $\sigma$-compact set $C\subset\mathbb R$ such that $C\cap X=A$; $\bullet$ a strong $Q$-set …

Is the following statement consistent? $(\star)$ Let $K$ be a separable compact Hausdorff space containing the space $[0,\omega_1]$ so that the complement $K\setminus[0,\omega_1]$ is discrete. Then t …

In fact, the MO user @trutheality was right saying that the cardinal $\kappa$ is equal to the continuum. This follows from Theorem. For any set $X\subset\mathbb R^d$ of cardinality $<\mathfrak c$ in …

Finally, PFA implies $\mathfrak P=\aleph_2=\mathfrak c=\mathfrak p>\aleph_1$, see Corollary 4.6 in Baumgartner's survey "Applications of the Proper Forcing Axiom" in the "Handbook of Set-Theoretic Topology …

In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic. By a partition …

The cardinal $\mathfrak{ridiculous}$ is equal to $\mathfrak p$ (which is equal to the smallest character of a free filter without infinite pseudointesection on $\omega$). It suffices to prove that a f …

Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on $T$-ultrafilters hoping that somebody on MO can solve it. First I recall the necessary defini …

Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is …

A topological space has countable spread if every discrete subspace is at most countable. By Theorem 8.10 in Todorcevic's book "Partition Problems in Topology", PFA implies that each regular space $X …

A function $f:\omega\to\omega$ is called $\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$; $\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite. Let us introduce …

Let $(P,\le)$ be a poset. For a point $x\in P$ let $${\downarrow}x=\{p\in P:p\le x\}\quad\text{and}\quad{\uparrow}x=\{p\in P:x\le p\}$$be the lower and upper sets of the point $x$, and for a subset $ …

At the moment we have the following information on the cofinalities of the poset $\mathfrak P$ (see Theorem 7.1 in this preprint). Theorem. 1) ${\downarrow}\!{\uparrow}\!{\downarrow}(\mathfra …