Question

2. Let (X, dx), (Y, dy), (2, dz) be metric spaces, and f : XY,g:Y + Z continu- ous maps. (a) Prove that the composition go f is continuous. (b) Prove that if W X is connected, then f(W) CY is connected.

Answer 1

2 Answer Let, (x, dx), (Y, dy), (2, dz) be three a ) Metric space and fix-YI:Y-1 z are Continuous Also, gof: x-z is Their composition Now, gf Wcz is ofen Then v-g' (W) is so, Š (v) is open it follows that (gof)(w) of (9' cw)) is open. open so so gof is continuous

We prove This result by contradiction. of few) is not Connected in Y, I non- empty disjoint subsets vv of f(w), which are open in few) such that f(w) 2 V uv AS v, v open in few) 3 an V, V iny such that v=un f(w) 3 Vavin f(w) are ofen sels By continuity of f f (0) & and f'(o) 2 (Vidow, f'(vi) are - open in f(wi) 2 f'(vidno X 2 Now few) 2 vuv w → - [s' (u) u f cu)] nw = [f're nw] o [ f(u) nw] f'( wil n N o [previ nw] Thus has been expressed as union of two sets which are open in W. Now, uintew) + 8 = f'(U)nw to

Similarly, f (V) on 70 Also, Unnin few) (= unu)=0 in [5'(U) ow]n[ f '(vi) n w) 29 W becomes dis connected a contradiction Therefore, of W cx is connected Then few) cy is connected.