### A Boolean view of sequential compactness

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We answer a question of I. Juhasz by showing that MA $+\neg $ CH does not imply that every compact ccc space of countable $\pi $-character is separable. The space constructed has the additional property that it does not map continuously onto ${I}^{{\omega}_{1}}$.

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.

The main goal of this paper is to establish a technical result, which provides an algorithm to prove several cardinal inequalities and relative versions of cardinal inequalities related to the well-known Arhangel’skii’s inequality: If $X$ is a ${T}_{2}$-space, then $\left|X\right|\le {2}^{L\left(X\right)\chi \left(X\right)}$. Moreover, we will show relative versions of three well-known cardinal inequalities.

The class of $s$-spaces is studied in detail. It includes, in particular, all Čech-complete spaces, Lindelöf $p$-spaces, metrizable spaces with the weight $\le {2}^{\omega}$, but countable non-metrizable spaces and some metrizable spaces are not in it. It is shown that $s$-spaces are in a duality with Lindelöf $\Sigma $-spaces: $X$ is an $s$-space if and only if some (every) remainder of $X$ in a compactification is a Lindelöf $\Sigma $-space [Arhangel’skii A.V., Remainders of metrizable and close to metrizable spaces, Fund. Math. 220 (2013),...

We establish a general technical result, which provides an algorithm to prove cardinal inequalities and relative versions of cardinal inequalities.

Relative versions of many important theorems on cardinal invariants of topological spaces are formulated and proved on the basis of a general technical result, which provides an algorithm for such proofs. New relative cardinal invariants are defined, and open problems are discussed.

We prove that ${\u2666}^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality ${2}^{{\aleph}_{1}}$ which has points ${G}_{\delta}$. In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

A condensation is a one-to-one continuous mapping onto. It is shown that the space ${C}_{p}\left(X\right)$ of real-valued continuous functions on $X$ in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum $X$ (Theorem 19). However, there exists a non-metrizable compactum $X$ such that ${C}_{p}\left(X\right)$ condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.

We give several partial positive answers to a question of Juhász and Szentmiklóssy regarding the minimum number of discrete sets required to cover a compact space. We study the relationship between the size of discrete sets, free sequences and their closures with the cardinality of a Hausdorff space, improving known results in the literature.