Chapter VII: Topology of the plane
Eduard Čech, Miroslav Katětov · Czech digital mathematics library · 1969
TOPOLOGY OF THE PLANE § 26.Cutting of the plane by a given set 26.1.In the topological study of the plane, a transfer from the plane to the sphere by the so called stereographical projection is often convenient.The sphere is the space S 2 (see 17.10).We put (throughout the whole chapter) co = (1,0, 0) g S 2 .If x 4-\y e E 2 , we put (throughout the whole chapter) o(x 4-iy) = (£0, > £2) e E S 2 -(co), whereBy the proof of theorem 17.10.4we obtain a is homeomorphic mapping of the plane onto S 2 -(co).The mapping a is termed the stereographical projection.The following theorem is easy to prove:The set M is unbounded if and only if coe a(M). Let MCEProof: By 26.1.1,o(M) is the closure of a(M) in S 2 -(co), so that 26.1.3follows from 8.7.1 and 26.1.2.26.1.4.Let M cz E 2 , ae M. A continuum K c (M) u (co) containing both o(a) and co exists if and only if there is a set C cz M which is closed (in E 2 ), connected, unbounded, and which contains the point a.Proof: I. Let C exist.The set o(C) a G(M) is connected by 26.1.1,so that, by 18.1.6,the set K = a(C) is also connected.We have K = o(C) u (co) by 26.1.3,so that K cz o(M) u (co), o(a) e K, (oeK.K is a continuum by 17.2.2 and 17.10.2.II.Let K exist.By 19.4.1 there exists an irreducible continuum L cz K between c(a) and co.Put Q = L -(co), C = cr-^0, SO that aeC, C