Concerning ordered spaces

Ordered spaces are an abstraction of the real line. This paper shows in Chapter 1 that all ordered spaces are hereditarily normal. In Chapter 2, necessary and sufficient conditions are given for a separable ordered space to be completely separable, and hence metrizable. In semi-metrizable ordered spaces the following are shown to be equivalent: (a) The space is completely separable. (b) The space is separable. (c) The space is hereditarily separable. (d) The space has the Lindelof property. (e) If M is an uncountable subset of the space, then some point of M is a limit point of M. (f) There does not exist an uncountable collection of mutually exclusive open sets in the space.