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Suppose that each set

X

of points in the plane has an associated set

\overline{X}

of points called its cover. Suppose further that (1)

\overline{X\cup Y} \supset \overline{\overline{X}} \cup \overline{Y} \cup Y

for all sets

X,Y

. Show that i)

\overline{X} \supset X

, ii)

\overline{\overline{X}}=\overline{X}

and iii)

X\supset Y \Rightarrow \overline{X} \supset \overline{Y}.

Prove also that these three statements imply (1).

B2