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Math · Advanced Math
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Answer all items of the following problem: Consider the closed unit interval0,1] with its usual topology ui, see Definition 5.4, page 33. Let J- [2, 3]. Define fJby()+2. Observe that f is a bijective function. Let X = 1 U J. . For each r e J, let 8()-ttr]). For each z I, let B(x) (wu (fw)\f()): eWeU For example, 0 e 1 and 0 E [0, eWi-Since f([0, ) = [2, 2) and f(0) = 2, then (o, j)u(I2,21)1 (2)) T, (2, 24) B0). Another example, consider I. We have feほ3 e ul. Also f((4,訒= ( 2, 2 ) and f()-21 Thus (,)U(f((, ))f))) ((21, 21)U(2,2) is a basic open neighborhood of That is

3. Prove or disprove: (X, T) is first countabie.

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