Approximating residual sets by strongly residual sets
Author:
D. A. Moran
Journal:
Proc. Amer. Math. Soc. 25 (1970), 752-754
MSC:
Primary 54.78
DOI:
https://doi.org/10.1090/S0002-9939-1970-0263053-0
MathSciNet review:
0263053
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $M$ be a closed topological manifold, $R$ residual in $M$, and $N$ any neighborhood of $R$ in $M$. The fulfillment by $R$ of a certain local separation property in $M$ implies that there exists a topological spine $Rβ$ of $M$ such that $N \supset Rβ \supset R$. (Topological spine = strongly residual set.) This local separation property is satisfied whenever $R$ is an $\operatorname {ANR}$, or when $\dim R \leqq \dim M - 2$.
- Morton Brown, A mapping theorem for untriangulated manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 92β94. MR 0158374
- P. H. Doyle and J. G. Hocking, A decomposition theorem for $n$-dimensional manifolds, Proc. Amer. Math. Soc. 13 (1962), 469β471. MR 141101, DOI https://doi.org/10.1090/S0002-9939-1962-0141101-1
- D. A. Moran, A remark on the Brown-Casler mapping theorem, Proc. Amer. Math. Soc. 18 (1967), 1078. MR 224076, DOI https://doi.org/10.1090/S0002-9939-1967-0224076-0
- R. H. Bing, Retractions onto ${\rm ANRβs}$, Proc. Amer. Math. Soc. 21 (1969), 618β620. MR 239583, DOI https://doi.org/10.1090/S0002-9939-1969-0239583-6
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Additional Information
Keywords:
Topological manifold,
residual set,
strongly residual,
topological spine,
Brown-Casler map
Article copyright:
© Copyright 1970
American Mathematical Society