Sub-posets in ωω and the strong Pytkeev⁎ property

Abstract

Tukey order is used to compare the cofinal complexity of partially order sets (posets). We prove that there is a 2c-sized collection of sub-posets in 2ω which forms an antichain in the sense of Tukey ordering. Using the fact that any boundedly-complete sub-poset of ωω is a Tukey quotient of ωω, we answer two open questions published in [14].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.