On countably compact, locally countable spaces

Almost compatible functions and infinite length games

“…A topological space is said to be ω-bounded if each countable subset of the space has compact closure. As in [6] we call a T 2 , locally countable, ω-bounded space splendid, and let S(κ) represent the claim that there exists a splendid space of cardinality κ. Alternatively, the previous corollary may be obtained via an observation of Todorcevic communicated by Dow in [3]: if every Kurepa family of size at most κ extends to a cofinal Kurepa family, then the same is true of κ + .…”

“…Nyikos points out in [9] that a cofinal Kurepa family may be used to construct a locally metrizable, ω-bounded, zero-dimensional space with appropriate cardinality, but whether this can be strengthened to locally countable and ω-bounded (as asked in [6]) remains an open question.…”

“…Also left open is this extension of the question asked in [9] and [6] on the possible equivalence of S(κ) and K(κ).…”

First countable, countably compact, noncompact spaces

“…had not yet heard of the joint work of Juhász, Nagy and Weiss [6] which produced arbitrarily large splendid spaces, which are more than enough for a consistent No answer to Problem 5. We now know that their construction works under e.g., Covering(V, K); for details see the article by Juhasz [5].…”

The Number of Closed Subsets of a Topological Space

“…Call a space good if it is regular, countably compact, and locally countable (i.e., every point has a countable open neighborhood). The following two results about good spaces are proved in [19]. (1) If X is an uncountable good space, then cf(\X\) > Ko- (2) If the axiom of constructibility holds, and cf(m) > Ko, then there is a good space of cardinality m. Now let X be a good space with \X\ = 2 Ku .…”