Sub-posets in $ω^ω$ and the Strong Pytkeev$^\ast$ Property

Abstract

Tukey order are used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2 c -sized collection of subposets in 2 ω which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ω ω is a Tukey quotient of ω ω , we answer two open questions published in [12].The relation between P -base and strong Pytkeev * property is investigated. Let P be a poset equipped with a second-countable topology in which every convergent sequence is bounded. Then we prove that any topological space with a P -base has the strong Pytkeev * property. Furthermore, we prove that every uncountably-dimensional locally convex space (lcs) with a P -base contains an infinite-dimensional metrizable compact subspace. Examples in function spaces are given.