  We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer
and Saturation Principles} in the framework of the superstructure of a given
infinite set. We also present several applications of this axiomatic approach
to point-set topology. Some of the topological topics such as the Hewitt
realcompactification and the nonstandard characterization of the sober spaces
seem to be new in the literature on nonstandard analysis. Others have already
close counterparts but they are presented here with essential simplifications.
