On continuously Urysohn and strongly separating spaces

A topological space X is continuously Urysohn if for each pair of distinct points x,y∈X there is a continuous real-valued function fx,y∈C(X) such that fx,y(x)≠fx,y(y) and the correspondence (x,y)→fx,y is a continuous function from X2⧹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 fx,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 fx,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 gx such that for any z∈X, gx(x)=gx(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 fx,y cannot be chosen independently of y. This answers a question raised by David Lutzer.