Strong monotone and nested normality by Robert E. Buck, Robert W. Heath and Phillip L. Zenor

Topology Proceedings 26 No. 1 (2001-2002), pp. 67-82

Strong monotone and nested normality

Robert E. Buck, Robert W. Heath and Phillip L. Zenor

Possible strengthenings of monotone normality are examined. Strong monotone normality: For each pair (H, K) of disjoint closed sets in X there is an open set G(H, K) such that (i.) H subset G(H, K) subset [`G(H, K] subset X \K; (ii.) for disjoint closed sets H^' and K^' with H subset H^' and K^' subset K, GH, K) GH^', K^'); (iii.) G(H, K) \cap G(K, H) = \phi; (iv.) if H^' is a closed set and H^' subset G(H, K), then G(H^', K) subset G(H, K). Nested normality requires properties (i.), (ii.), and (iv.). Metric spaces, compact monotonically normal spaces, and linearly ordered topological spaces are among the strongly monotonically normal spaces. Strong monotone normality is preserved by closed maps among other things. Using the fact that every stratifiable space can be embedded (as a perfect retract) in an M₁-space, one can prove that if every M₁-space is strongly monotonically normal, then so is every stratifiable space. It is not known whether every M₁-space is strongly monotonically normal. A version of the Dugundji Extension Theorem is proved for nestedly normal spaces.

Keywords: acyclic monotone normality, adjunction spaces, extension theorems, LOTS, metric space, monotone normality, monotone T₂, nested normality, proto-metrizable, retract, stratifiable, strong monotone normality, strong monotone T₂
Mathematics Subject Classification: 54B05, 54B17, 54C20, 54C55, 54D15, 54E20

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Topology Proceedings, Volume 26 No. 1 (2001-2002)
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