Q7 please

Examples . :1 Any finite Space is Compact. 2 ( Since the ne of possible subsets of it is finite) so the over itself is finite Theis any finite subset of any top, space is compact, 2 ( X. ) is always compact: 7) finit - compact 3) If (X is infinite), then (X This ) is never compact, since flag:sexy is an open cover of X with no finite subcover. 4) (RT) is not compact. For; $=[(NIN): NEN's is an open cover of R. ( justify ) If EAN sun- Ann's is a finite subsets of , then let Me= max Enismighty . Now ne $UAn . Hence such a collection does not cover R. K = 1 In (RTs) , A= Cojustineng is compact. If is an open cover of A, then choose Here with ofl. The set u contains all but finitely many of the points of the set Et: new} in the sense that 3 NENI TELL forall ny N. For each ISAKIN, choose UnEND with A Elly. Then BElouis inIs Ung is a finite subcover of of. A = (091 ] In (Rots ), is not compact If = ECT,IT: nENG is an open cover of A which has no finite subcover. Also (091) is not compact. [-lol] is compact in (Ronks ) IN is not compact in (RST ) since in is the discrete top and the space under the disc. top."is compact iff it is finite. Scanned with CamScanner