### A construction of the projective modification for a closure-set of a presheaf

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The completion of a Suslin tree is shown to be a consistent example of a Corson compact L-space when endowed with the coarse wedge topology. The example has the further properties of being zero-dimensional and monotonically normal.

A constructively valid counterpart to Bourbaki’s Fixpoint Lemma for chain-complete partially ordered sets is presented to obtain a condition for one closure system in a complete lattice $L$ to be stable under another closure operator of $L$. This is then used to deal with coproducts and other aspects of frames.

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...

The Katětov ordering of two maximal almost disjoint (MAD) families $\mathcal{A}$ and $\mathcal{B}$ is defined as follows: We say that $\mathcal{A}{\le}_{K}\mathcal{B}$ if there is a function $f:\omega \to \omega $ such that ${f}^{-1}\left(A\right)\in \mathcal{I}\left(\mathcal{B}\right)$ for every $A\in \mathcal{I}\left(\mathcal{A}\right)$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in \mathcal{I}{\left(\mathcal{A}\right)}^{+}$, we have that ${\mathcal{A}|}_{X}{\le}_{K}\mathcal{A}$. We prove that CH implies that for every $K$-uniform MAD family $\mathcal{A}$ there is a $P$-point $p$ of ${\omega}^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense in the...

We consider the question of when ${X}_{M}=X$, where ${X}_{M}$ is the elementary submodel topology on X ∩ M, especially in the case when ${X}_{M}$ is compact.

We characterize the subsets of the Alexandroff duplicate which have a G${}_{\delta}$-diagonal and the subsets which are M-spaces in the sense of Morita.