More on continuously Urysohn spaces

We study the properties of weakly continuously Urysohn and 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. By using the scattering process, we show that the class of protometrizable spaces is also contained in the class of continuously Urysohn space. We also give a characterization of the continuously Urysohn property for well-ordered spaces, and prove that a paracompact locally continuously Urysohn ordered space is continuously Urysohn.