Glasnik Matematicki, Vol. 52, No. 1 (2017), 185203.
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, 6578501,
Japan
email: tmiyata@kobeu.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 procategory
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) (access from subscribing institutions only)
DOI: 10.3336/gm.52.1.14
References:

N. K. Bilan,
Comparing monomorphisms and epimorphisms in pro and pro^{*}categories,
Topol. Appl. 155 (2008), 18401851.
MathSciNet
CrossRef

J. Dydak and F. R. Ruiz del Portal,
Monomorphisms and epimorphisms in procategories,
Topol. Appl. 154 (2007), 22042222.
MathSciNet
CrossRef

T. Miyata and T. Watanabe,
Approximate resolutions of the fractal category,
Glas. Mat. Ser. III 38 (2003), 377393.
MathSciNet
CrossRef

S. Mardešić,
Approximate polyhedra, resolutions of maps and shape fibrations,
Fund. Math. 114 (1981), 5378.
MathSciNet

S. Mardešić and L. Rubin,
Approximate inverse systems of compacta and covering dimension,
Pacific J. Math. 138 (1989), 129144.
MathSciNet
CrossRef

S. Mardešić and J. Segal,
Shape Theory,
NorthHolland, AmsterdamNew York, 1982.
MathSciNet

S. Mardešić and J. Segal,
Mapping approximate inverse systems of compacta,
Fund. Math. 134 (1990), 7391.
MathSciNet

S. Mardešić and T. Watanabe,
Approximate resolutions of spaces and mappings,
Glas. Mat. Ser. III 24 (1989), 587637.
MathSciNet

T. Watanabe, Approximative shape I,
Tsukuba J. Math. 11 (1987), 1759.
MathSciNet
CrossRef
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