Exercise 8, § 4

Let $\mathfrak{R} = (\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ and $\mathfrak{S} = (\mathrm{Y}_\varkappa)_{\varkappa \in \mathrm{K}}$ be two coverings of a set E.

(a) Show that if $\mathfrak{R}$ and $\mathfrak{S}$ are partitions of E and if $\mathfrak{R}$ is finer than $\mathfrak{S}$, then for every $\varkappa \in \mathrm{K}$ there exists $\iota \in \mathrm{I}$ such that $\mathrm{X}_\iota \subset \mathrm{Y}_\varkappa$.

(b) Give an example of two coverings $\mathfrak{R}$, $\mathfrak{S}$ such that $\mathfrak{R}$ is finer than $\mathfrak{S}$ but such that the property stated in (a) is not satisfied.

(c) Give an example of two partitions $\mathfrak{R}$, $\mathfrak{S}$ of E such that for every $\varkappa \in \mathrm{K}$ there exists $\iota \in \mathrm{I}$ such that $\mathrm{X}_\iota \subset \mathrm{Y}_\varkappa$ but such that $\mathfrak{R}$ is not a refinement of $\mathfrak{S}$.