Many fundamental mathematical results fail in ZF, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results — old and new — that specify how much “choice” is needed precisely to validate each of certain basic analytical and topological results.

In this paper we show that a separable space cannot include closed discrete subsets which have the cardinality of the continuum and satisfy relative versions of any of the following topological properties: normality, countable paracompactness and property $\left(a\right)$. It follows that it is consistent that closed discrete subsets of a separable space $X$ which are also relatively normal (relatively countably paracompact, relatively $\left(a\right)$) in $X$ are necessarily countable. There are, however, consistent examples of...

Let us denote by $\Phi (\lambda ,\mu )$ the statement that $𝔹\left(\lambda \right)=D{\left(\lambda \right)}^{\omega }$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $ℂ$ in $𝔹\left(\lambda \right)$ picks up all the $\mu$ colors. We call a space $X$$\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V}\subset U$. We recall that a space $X$ is called feebly compact if...

This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width.

Let $\Psi \left(\Sigma \right)$ be the Isbell-Mr’owka space associated to the $MAD$-family $\Sigma$. We show that if $G$ is a countable subgroup of the group $𝐒\left(\omega \right)$ of all permutations of $\omega$, then there is a $MAD$-family $\Sigma$ such that every $f\in G$ can be extended to an autohomeomorphism of $\Psi \left(\Sigma \right)$. For a $MAD$-family $\Sigma$, we set $Inv\left(\Sigma \right)=\{f\in 𝐒(\omega ):f[A]\in \Sigma$ for all $A\in \Sigma \}$. It is shown that for every $f\in 𝐒\left(\omega \right)$ there is a $MAD$-family $\Sigma$ such that $f\in Inv\left(\Sigma \right)$. As a consequence of this result we have that there is a $MAD$-family $\Sigma$ such that $n+A\in \Sigma$ whenever $A\in \Sigma$ and $n<\omega$, where $n+A=\{n+a:a\in A\}$ for $n<\omega$. We also notice that there is no $MAD$-family $\Sigma$ such...

For $\varnothing \ne M\subseteq {\omega }^{*}$, we say that $X$ is quasi $M$-compact, if for every $f:\omega \to X$ there is $p\in M$ such that $\overline{f}\left(p\right)\in X$, where $\overline{f}$ is the Stone-Čech extension of $f$. In this context, a space $X$ is countably compact iff $X$ is quasi ${\omega }^{*}$-compact. If $X$ is quasi $M$-compact and $M$ is either finite or countable discrete in ${\omega }^{*}$, then all powers of $X$ are countably compact. Assuming $CH$, we give an example of a countable subset $M\subseteq {\omega }^{*}$ and a quasi $M$-compact space $X$ whose square is not countably compact, and show that in a model of A. Blass and S. Shelah every quasi...

We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.

A space X is called an α-Toronto space if X is scattered of Cantor-Bendixson rank α and is homeomorphic to each of its subspaces of the same rank. We answer a question of Steprāns by constructing a countable α-Toronto space for each α ≤ ω. We also construct consistent examples of countable α-Toronto spaces for each $\alpha <{\omega }_1$.

We present a construction from ♢* of a first countable, regular, countably metacompact space with a closed discrete subspace that is not a $G_{\delta }$. In addition some nonperfect spaces with σ-disjoint bases are constructed.

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ${}^{\omega }\omega$ of irrationals, or certain of its subspaces. In particular, given $f\in {}^{\omega }(\omega \setminus \left\{0\right\})$, we consider compact sets of the form ${\prod }_{i\in \omega }{B}_i$, where $|B_i|=f\left(i\right)$ for all, or for infinitely many, $i$. We also consider “$n$-splitting” compact sets, i.e., compact sets $K$ such that for any $f\in K$ and $i\in \omega$, $\left|\right\{g\left(i\right):g\in K,g\upharpoonright i=f\upharpoonright i\left\}\right|=n$.

We show that it is consistent with ZF that there is a dense-in-itself compact metric space $(X,d)$ which has the countable chain condition (ccc), but $X$ is neither separable nor second countable. It is also shown that $X$ has an open dense subspace which is not paracompact and that in ZF the Principle of Dependent Choice, DC, does not imply the disjoint union of metrizable spaces is normal.

On ${\omega }_1$ a Hausdorff inverse Lindelöf non Lindelöf topology has been constructed.

In [Fund. Math. 210 (2010), 1-46] we claimed the truth of two statements, one now known to be false and a second lacking a proof. In this "Errata" we report these matters in the interest of setting the record straight on the status of these claims.

Following Malykhin, we say that a space $X$ is extraresolvable if $X$ contains a family $𝒟$ of dense subsets such that $\left|𝒟\right|>\Delta \left(X\right)$ and the intersection of every two elements of $𝒟$ is nowhere dense, where $\Delta \left(X\right)=min\left\{\right|U|:U$ is a nonempty open subset of $X\}$ is the dispersion character of $X$. We show that, for every cardinal $\kappa$, there is a compact extraresolvable space of size and dispersion character $2^{\kappa }$. In connection with some cardinal inequalities, we prove the equivalence of the following statements: 1) $2^{\kappa }<2^{{\kappa }^{+}}$, 2) ${\left({\kappa }^{+}\right)}^{\kappa }$ is extraresolvable and...