An Introduction to Classical Complex Analysis: 1 by Robert B. Burckel

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By Robert B. Burckel

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This will be essential to the whole Cauchy theory of holomorphic functions and also will be one of the main tools in the rather complete topological analysis of the plane offered in Chapter IV. The presentation here, while maximally concise, has a deus ex machina character. ” One has only to show somehow, without the exponential, that these two groups are locally isomorphic. Similarly with the real exponential x --f ex. This elegant, but arduous, program is carried out in BOURBAKI [1966], chapter V, Q 3 and 4, and chapter VIII, 9 2.

Such exotica indeed exist; see page 145 of STEENand SEEBACH [1970]. We may situate Cin C, so that x = (0,0,l). Then let S = Q:,\C, a subset of C,\{(O,O, 1)) = C. Then of course this set has a connected complement in C , but its complement in C is the totally disconnected set C\{(O, 0, I)} and so has infinitely many bounded components. 14 is true: any two points on any curve lie on an arc which lies on the curve. For a proof of this surprisingly non-trivial fact (due to R. L. MOORE[1916/17]) see TIETZE[1919], MAZURKIEWICZ [1920], KAMIYA [1931] or G.

30 Let X be a non-empty subset of @. Then Each component of X is relatively closed. (i) (ii) Distinct components are disjoint. (iii) Each connected subset of X lies in a unique component. (iv) X is the (disjoint) union of its components. If X is open, then: (i)’ Each component of X is open and there are at most countably many of tltem. 2). (ii) If C, and C, are components with a point in common, then C , u C , is connected by I . 1 and so by the maximality of C, and C, we see that c, = c, u c, = c,.

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