### On a Corson space of Todorčević

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Under the assumption that the real line cannot be covered by ${\omega}_{1}$-many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into ${\omega}_{1}$-many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into ${\omega}_{1}$-many closed sets; and (c) no compact Hausdorff space can be partitioned into ${\omega}_{1}$-many closed ${G}_{\delta}$-sets.

We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel’skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in ${\left[\omega \right]}^{\omega}$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.

We show that MA${}_{\sigma -centered}\left({\omega}_{1}\right)$ implies that normal locally compact metacompact spaces are paracompact, and that MA(${\omega}_{1}$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.

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 $\alpha <{\omega}_{1}$.

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 introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Lindelöf spaces are shown.

We show that if $X$ is a subspace of a linearly ordered space, then ${C}_{k}\left(X\right)$ is a Baire space if and only if ${C}_{k}\left(X\right)$ is Choquet iff $X$ has the Moving Off Property.

For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an ${L}_{ch}$-extender (resp. ${L}_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X; (iii) there is a continuous ${L}_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and...

Natural weakenings of uniformizability of a ladder system on ω₁ are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The...

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