On continuously Urysohn and strongly separating spaces

Abstract

A topological space X is continuously Urysohn if for each pair of distinct points x, y∈ X there is a continuous real-valued function f x, y ∈ C( X) such that f x, y ( x)≠ f x, y ( y) and the correspondence ( x, y)→ f x, y is a continuous function from X 2⧹ Δ to C( X), where C( X) carries the topology of uniform convergence and Δ={(x,x): x∈X} . Metric spaces are examples of continuously Urysohn spaces with the additional property that the functions f x, y depend on just one parameter. We show that spaces with this property are precisely the spaces that have a weaker metric topology. However, to find an example of a continuously Urysohn space where the functions f x, y cannot be chosen independently of one of their parameters, it is easier to consider a much simpler property than “continuously Urysohn”, given by the following definition: A topological space X is strongly separating if for each point x∈ X there is a continuous, real-valued function g x such that for any z∈ X, g x ( x)= g x ( z) implies x= z. We show that a continuously Urysohn space may fail to be strongly separating. In particular, the example that we present is a continuously Urysohn space, where the Urysohn functions f x, y cannot be chosen independently of y. This answers a question raised by David Lutzer.