(solution) Let S be a nonempty subset of R such that if H1 is any open

(solution) Let S be a nonempty subset of R such that if H1 is any open

Let S be a nonempty subset of R such that

if H1 is any open covering of S, then S has an open sub-

covering H???? of finitely many open sets from H. Show that

S is compact. (Hint : First show that S is bounded by

considering the open covering H1 = {(s?1, s+1)|s ? S}

of S. To show that S is closed, proceed by contradiction :

assume S is not closed, then there exists a limit point

s0 ?/ S. Now consider the open covering of S given by

H2 = {(s ? |s?s0|,s + |s?s0|)|s ? S}, and show that no 

sub-covering of H2 covers S.)