Question

2. Prove the following Theorems: (a). Prove that the real line with the standard topology is Hausdorff. (b). Prove that int(ANB) = int(A) n int(B) Y is a homeomorphism. Then if X is a (c). If X and Y are topological spaces and f: X Hausdorff space then Y is Hausdorff. (d). Theorem 4.2

Answer 1

2a) Prove that the meal Gne with the stamdamd is taurdnoff topog Proff det, without amd e IR Such that x LACK of gemamality We assunme that Ket, us chorse, e Them, (x-e, +e)n ca-t+) ve, intemsection of open sets of 2 and y are so, CIR, Tu)is Hausdre ff. b Prove that int (AnB) int (A) nint () Proff Simce, AABSA cmt CA) int CAOB) simi lanly AOBSB int CAOB)smt CB). nt (A) nmt C8) So int (A O8) int CA) aint CB) is am opem int CANB) In othem hand Set contaurn ed im AnB, amd s the Lamgest open set contained in An8. int CA)n int (B) int CANB) 2) 1 We fmom u) and 2) we have imt (AnB) mt CA) int CB) 12

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