### The space ${(\omega *)}^{n+1}$ is not always a continuous image of ${(\omega *)}^{n}$

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Some relatives of the Juhász Club Principle are introduced and studied in the presence of CH. In particular, it is shown that a slight strengthening of this principle implies the existence of a Suslin tree in the presence of CH.

The relativization of Gryzlov’s theorem about the size of compact ${T}_{1}$-spaces with countable pseudocharacter is false.

Galois-Tukey equivalence between matrix summability and absolute convergence of series is shown and an alternative characterization of rapid ultrafilters on ω is derived.

It is shown that no generalized Luzin space condenses onto the unit interval and that the discrete sum of ${\aleph}_{1}$ copies of the Cantor set consistently does not condense onto a connected compact space. This answers two questions from [2].

The authors give a ZFC example for a space with $\mathit{\text{Split}}\phantom{\rule{0.8pt}{0ex}}(\Omega ,\Omega )$ but not $\mathit{\text{Split}}\phantom{\rule{0.8pt}{0ex}}(\Lambda ,\Lambda )$.

Players ONE and TWO play the following game: In the nth inning ONE chooses a set ${O}_{n}$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset ${T}_{n}$ of X. The players must obey the rule that ${O}_{n}\subseteq {O}_{n+1}\subseteq {T}_{n+1}\subseteq {T}_{n}$ for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a ${G}_{\delta}$-set. To what extent is the converse true? We show that: (A) For ℱ the collection of countable subsets of X: 1. There are subsets...

We study conditions under which sequentially continuous functions on topological spaces and sequentially continuous homomorphisms of topological groups are continuous.

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