abstract
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On continuously Urysohn and strongly separating spaces

Lorenz Halbeisen and Norbert Hungerbühler

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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.