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: XY 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.


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DOI: 10.3336/gm.52.1.14


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