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$ C$-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
MathSciNet review: 0328882
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Abstract: A space $ X$ has Property $ {\text{C}}$ provided that for every separated closed subset $ A$ of $ X$ there exist disjoint closed and connected sets $ {C_1}$ and $ {C_2}$ in $ X$ each of which intersects $ A$ and which contain $ A$ in their union. This property has been used in characterizing unicoherence. A metric space $ X$ has Property $ {\text{S}}$ if for each $ \varepsilon > 0,X$ is the union of a finite number of connected sets each of diameter less than $ \varepsilon $. In this paper a sufficient condition for a space to have Property $ {\text{C}}$ is established and used to show that separable Hilbert space has Property $ {\text{C}}$ and that every connected metric space having Property $ {\text{S}}$ has Property $ {\text{C}}$. It follows from the latter result that every separable, locally connected, connected, rimcompact metric space has Property $ {\text{C}}$. An example is given of a unicoherent, connected, uniformly locally connected, locally arcwise connected, separable metric space that does not have Property $ {\text{C}}$.

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Keywords: $ C$-separated, Property $ {\text{C}}$, Property $ {\text{S}}$, Hilbert space manifold
Article copyright: © Copyright 1973 American Mathematical Society

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