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Cxg nxx cu cxg. Proof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex · 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ N. We showed that /\(A c X, A is closed) /\(bX\A) \/(U,BG) (A c U and bB and UnB=O), which says that (X,G) is a T_3 space Theorem TSA11 If (X,G) is a topological space then the following conditions are equivalent (a) (X,G) is a T_4 space, (b) /\(UG) /\(A c U, A is closed) \/(EG) A c E and clo(E) c U Proof (a)=>(b) Take any open set. T C K i ~ b ` F c V ɑ啝 ɍ 蒼 u e җ v ƂȂ H.

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