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Nonstandard topology and extensions of monad systems to infinite points

Intuitively, a topological space is a set together with some notion of “nearness”. Classically, this notion of nearness is usually described by a collection of open sets. With the development of Non-standard analysis by Abraham Robinson [11], we have an alternative way to describe this notion. If X is any set and *X is a nonstandard extension of X then we can describe a topology on X by means of a relationship of “infinitely close” on some points of X. In many ways this latter approach is more intuitive and leads to more straightforward proofs [11] than the classical approach. The first section of this paper explores the connections between these two approaches. In the main section of the paper we extend the notion of “infinitely close” to infinite points of *X. In the final section we give a very intuitive characterization of the compact-open topology (when the domain is locally compact) and use it to give quick, straight-forward proofs of the usual facts about this topology [1], [2]. This last section can be read independently of the second section.