Glasnik Matematicki, Vol. 52, No. 1 (2017), 185-203.

APPROXIMATE MAPS CHARACTERIZING INJECTIVITY AND SURJECTIVITY OF MAPS

Takahisa Miyata

Department of Mathematics and Informatics, Graduate School of Human Development and Environment, Kobe University, Kobe, 657-8501, Japan
e-mail: tmiyata@kobe-u.ac.jp

Abstract.   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.

2010 Mathematics Subject Classification.   54C56, 54C25, 54B30.

Key words and phrases.   Injective map, surjective map, epimorphism, monomorphism, approximate map, shape.

Full text (PDF) (free access)

DOI: 10.3336/gm.52.1.14

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