Let S be a subset of R. Mark each statement True or False. Justify each answer.?

(a) If S is not open, then S is closed.


(b) There can鈥檛 be a set which is both open and closed.


(c) Every isolated point of S is a boundary point of S.


(d) S is closed iff S contains its boundary points.


(e) S is closed iff S contains its accumulation points.


(f) If S is compact, then S is closed.


(g) If S is closed, then S is compact.


(h) The intersection of any collection of open sets is open.


(i) If intS (the set of the interior points of S is empty, then bdS =S,


where bdS is the set of the boundary points of S.Let S be a subset of R. Mark each statement True or False. Justify each answer.?
(a) false, example [0;1[ in R


(b) false, R is open and closed in R


(c) true, if x is isolated in S than x is in S and so in clos(S) and not in int(S). So x is a member of bd S = clos(S) \ int(S)


(d) true


(e) true


(f) true, In R we have: S is compact, iff S is closed and bounded.


(g) false, S has to be bounded and closed


(h) false, Intersection of ]-1/n ; 1/n [ for all n is {0}


(it's true for finite number of open sets)


(i) false, int({1/n | n positive integer} is empty, but 0 belongs to the boundary and not to S.Let S be a subset of R. Mark each statement True or False. Justify each answer.?
a. true


b. true


h. false