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Absolute Approximate Retracts and AR-Spaces

Published online by Cambridge University Press:  20 November 2018

John R. Martin
John R. Martin*
Affiliation:
University of Saskatchewan, Saskatoon, Saskatchewan
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A subset A of a topological space X is an approximateretract of X if for every neighborhood U of A in X there is a retract R of X such that A ⊂ R ⊂ U. A compactum X is an absolute approximate retract (AAR-space) if whenever X is embedded as a subset of a compactum Z, then X is an approximate retract of Z. These concepts were first defined in [2] where it is shown that every AAR-space is a contractible Peano continuum. In [3] an example is given to show that there exists a contractible LC∞ compactum which is not an AAR-space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 297 - 301
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Borsuk, K., Theory of retracts, Monografie Matema t y cz ne 44 (Warszawa, 1967).Google Scholar
2. Martin, J. R., A generalization of absolute retracts, Proc. Amer. Math. Soc. 52 (1975), 409–413.Google Scholar
3. Martin, J. R., An example of a contractible LC°° compactant which is not an absolute approximate retract, Bull. Ac. Pol. Sri. 25 (1977), 489–492.Google Scholar
4. Martin, J. R., Absolute fixed point sets and AR-spaces, Fund. Math, (to appear).Google Scholar
5. Martin, J. R., Absolute fixed point sets in compacta, Colloq. Math. 39 (1978), 41–44.Google Scholar