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Question: answer all items of the following problem consider the closed...

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Answer all items of the following problem Consider the closed unit interval I = [0, 1] with its usual topology ur, see Definition 5.4, page 33. Let J 2,3]. DefinefIJ by f(a)+ Observe that f is a bijective function. Let X = IUJ. For each E J, let 93(z) = {(c)) For each r e 1, let (z) = { W U (f(W) \ {f(z)) ) : r e w e uh For example, 0 E 1 and 0 e [0. ) E ur. Since f(0, )) = [2 , 2) and f (0-2, then (o, u(,2 2)) , (2,23) B0) Another example, consider e 1. We have ,訇e ur. Also /((hj)) = (21 , 2.1 ) and f() = 24. Thus( , j) U(f(( 3 3 ))\ {f( ))) ( :. )U( 21, 2 ) ( .2 . ) is a base open neighborhood of . That is, (u (2,21)U (2, 2) B).

7. Prove that (X, T) is T2.

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