Let U A X and suppose that U is open in X Show that U clA cl

Let U, A X and suppose that U is open in X. Show that U cl(A) cl (U A), and cl(U cl(A)) = cl(U A).

Solution

A set is closed iff its complement is open.

The intersection of two open sets is open.

AA is closed in XX XAXA is open in XX

UU is open in XX

So (XA)U(XA)U is open in XX ( finite intersection of open sets is open)

(XU)(AU)(XU)(AU) is open in XX

U(AU)U(AU) = UAUA (can be seen using the venn-diagram)

UAUA is open in XX

similarly for AUAU

XUXU is closed in XX

A(XU)A(XU) is closed in XX ( arbitrary union of closed sets is closed)

= (AX)(AU)=A(AU)=AU(AX)(AU)=A(AU)=AU

Hence, AUAU is closed in X.

A set is closed iff its complement is open. The intersection of two open sets is open.