-separated sets in certain metric spaces

Authors:
R. F. Dickman, R. A. McCoy and L. R. Rubin

Journal:
Proc. Amer. Math. Soc. **40** (1973), 285-290

MSC:
Primary 54F15

DOI:
https://doi.org/10.1090/S0002-9939-1973-0328882-6

MathSciNet review:
0328882

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Abstract: A space has Property provided that for every separated closed subset of there exist disjoint closed and connected sets and in each of which intersects and which contain in their union. This property has been used in characterizing unicoherence. A metric space has Property if for each is the union of a finite number of connected sets each of diameter less than . In this paper a sufficient condition for a space to have Property is established and used to show that separable Hilbert space has Property and that every connected metric space having Property has Property . It follows from the latter result that every separable, locally connected, connected, rimcompact metric space has Property . An example is given of a unicoherent, connected, uniformly locally connected, locally arcwise connected, separable metric space that does not have Property .

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0328882-6

Keywords:
-separated,
Property ,
Property ,
Hilbert space manifold

Article copyright:
© Copyright 1973
American Mathematical Society