On continuously Urysohn and strongly separating spaces

A topological space X is continuously Urysohn if for each pair of distinct points x,y in X there is a continuous real-valued function f_x,y in C(X) such that f_x,y(x) is not equal to f_x,y(y) and the correspondence (x,y) to f_x,y is a continuous function from X × X minus the diagonal to C(X), where C(X) carries the topology of uniform convergence. 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 in X there is a continuous, real-valued function g_x such that for any z in 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.

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