A note on paracompactness in generalized ordered spaces

Abstract

In this paper, we show that for generalized ordered spaces, paracompactness is equivalent to Property D, where a space X X is said to have Property D if, given any collection { G ( x ) : x ∈ X } \{G(x)\colon x\in X\} of open sets in X X satisfying x ∈ G ( x ) x\in G(x) for each x x , there is a closed discrete subset D D of X X satisfying X = ⋃ { G ( x ) : x ∈ D } X=\bigcup \{G(x)\colon x\in D\} .