Approximate Maps Characterizing Injectivity and Surjectivity of Maps

In the theory of inverse systems, in order to study the properties of a space X or a map f : X → Y between spaces, one expands X to an inverse system X or expands f to a map f : X → Y between the inverse systems, and then work on X or f . In this paper, we define approximate injectivity (resp., surjectivity) for approximate maps, and show that a map f : X → Y between compact metric spaces is injective (resp., surjective) if and only if any approximate map f : X → Y whose limit is f is injective (resp., surjective). As a consequence, we show that an approximate map f : X → Y is approximately injective (resp., approximately surjective) if and only if f represents a monomorphism (resp., an epimorphism) in the approximate pro-category in the sense of Mardešić and Watanabe.