Sergio Salbany, Todor Todorov
Nonstandard Analysis in Point-Set Topology
Preprint series: ESI preprints

MSC:

03H05 Nonstandard models in mathematics, See also {26E35, 28E05, 30G06, 46S20, 47S20, 54J05}

54J05 Nonstandard topology, See also {03H05}

54D10 Lower separation axioms, ($T_0$--$T_3$, etc.)

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

54D30 Compactness

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)

54D60 Realcompactness and realcompactification

Abstract: 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.
Keywords: nonstandard extension, monad, transfer principle, saturation principle, nonstandard hull, compactifications, realcompactification, separation axioms, T_0, T_1, T_2, T_3, T_4, compactness, soberness