On Continuously Urysohn Spaces

Abstract

We study the properties of weakly continuously Urysohn (denoted by wcU ) and continuously Urysohn (denoted by cU ) spaces. The class of continuously Urysohn spaces is known to contain the class of metrizable,submetrizable, and nonarchimedean spaces. In this work, by using the scattering process, we show that the class of proto-metrizable spaces is also contained in the class of continuously Urysohn spaces. We show that being a (weakly) continuously Urysohn space is not a multiplicative property, and that this property is not preserved under perfect maps. However, being a weakly continuously Urysohn space is preserved under perfect open maps. We give a proof that the topological sum of (weakly) continuously Urysohn spaces is also (weakly) continuously Urysohn and that any paracompact locally continuously Urysohn ordered space is also continuously Urysohn. We prove that a well-ordered space is continuously Urysohn if and only if it is hereditarily paracompact and we obtain a result which characterizes when the linear extension of a separable GO-space is continuously Urysohn.