Continuous separating families in ordered spaces and strong base conditions

Abstract

In this paper we study the role of Stepanova's continuous separating families in the class of linearly ordered and generalized ordered spaces and we construct examples of paracompact spaces that have strong base properties (such as point-countable bases or σ-disjoint bases), have continuous separating families, and yet are non-metrizable.

A Note on Continuously Urysohn Spaces

“…As separability is finite multiplicative, X 2 is separable. Because X is a Hausdorff space, is closed in X and hence X 2 \ is an open subset of X 2 . We know that separability is hereditary with respect to open subsets, so X 2 \ is also separable.…”

“…Then, applying Theorem 2.1, we know that X has a weaker metric topology. Using [ [2]. They proved it only for generalized ordered spaces.…”

A NOTE ON PARACOMPACT p-SPACES AND THE MONOTONE D-PROPERTY

“…Gao and W.-X. Shi [2] Let X be a generalized ordered (GO) space with the underlying linearly ordered topological space X u , let X * be the minimal closed linearly ordered extension of X andX be the minimal dense linearly ordered extension of X . Clearly, separability, countable chain condition and Lindelöfness of X can be preserved byX .…”

“…However, these properties of X are not hereditary toX (see [5]). Since the Sorgenfrey line S and the Michael line M are submetrizable they have the continuous Urysohn property P. M * has P [2], but S * andS do not have P [8].…”