Nearly metacompact spaces

A note on monotonically metacompact spaces

Article history: Received 17 July 2008 Received in revised form 6 October 2009 MSC: primary 54D20 secondary 54E30, 54E35, 54F05

Compact Monotonically Metacompact Spaces Are Metrizable

Monotonically metacompact spaces were recently introduced as an extension of the concept of monotonically compact spaces. In this note we answer a question of Popvassilev, and Bennett, Hart, and Lutzer, by showing that every compact, Hausdorff, monotonically (countably) metacompact space is metrizable. We also show that certain countable spaces fail to be monotonically (countably) metacompact.

Nearly Countable Dense Homogeneous Spaces

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X \ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of count...

Products of Nearly Compact Spaces

Nearly equal distances in metric spaces

Let (X, d) be any finite metric space with n elements. We show that there are two pairs of distinct elements in X that determine two nearly equal distances in the sense that their ratio differs from 1 by at most 9 logn n2 . This bound (apart for the multiplicative constant) is best possible and we construct a metric space that attains this bound. We discus related questions and consider in part...