$C$-separated sets in certain metric spaces

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 ${\text {C}}$.