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Math · Advanced Math
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Consider the closed unit interval 1 [0, 1] with its usual topology lr, see Definition 5.4, page 33. Let J [2, 31. Define f IJ by f(x)+2. Observe that f is a bijective function. Let X - IUJ. For each a E J, let B(x) (f For each z E 1, let 8(2) = { Wu ( f(W) \ {f(x)) ) : x E W E For example, 0 I and 0 e [o, ) Eu. Since f([o, )) [2, 21) and f(0) 2, then [o, U (12, 2)2))-o, )u(2,21) B(O) Another example, consider I. We have e (3) Eui Also f((3. )) (21, 23) and f() 2 Thus ( L((2,21)U (2, 2) is a basic open neighborhood of.at i,(2)2) Prove or disprove: (x, T) is first countabie.

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