Question

Only part (a).

(51) Michael's Line. Let S denote R with the topology generated by the base P(P) U {(a,b) b, a, b, E R} (see 2D(9)). Note that T(R) C T(S) where T(R) is the usual topology on R. a < (a) Show S is a completely Hausdorff space (use 5.29(a)) (b) Show S has a base of clopen sets and conclude that S is Tychonoff (see the comment before 5.13

denotes the power set of the irrationals.

Answer 1

different topology Also since the collection of sets {(a, b)la < b, a, bE R} is a basis for the usual R, and it is part of the basis set for the Michael Line topology, that the Micheal Line topology is finer than the usual topology. From the will get the Micheal Line topology to be completely hausdorff, if we get that R is completely hausdorff when equipped with the usual topology Now let r, y E R, with < y. Considerf R - [0, 1] defined as follows: We first note that S is the same set as R only with a topology on we see theorem we Vc r VICSY Vy c 0 C一電 f(e) 一エ Clearly f is the required continuous function that seperates the two points and y. (If you don't want to use the theorem, the same function works as it is also continuous on the Micheal Line topology)