C.1 The Long Line We follow the outline of Exercise 12 of §24. denote the set L Let denote the smallest element of Let o Lemma C.1. [o 0 X 0, ckX 0] interval S of has the order type of L 0 in 5S_. Sppose the lemma holds for all We show it holds for 3. . 4< [0,1]. i: the immediate successor Proof. Note that the proof is trivial if oL of Tren the o0. SLddifferent from bEx a point of ( the order topology. L Give . Let in the dictionary order. Stx [0,1), If has an immediate predecessor c1' the proof is easy. The interval [dOx 0, lX 0] of L has the order type of [0,1] by hypothesis. Tlhe interval l ; . [ has the order type of of L of [0,2], = <[0, 1)) L I equals ( and also of [0,1], [],2] [0,1] order type of 0 ,i+l 0] of [c,l], which of course has the order type of O] [0,1]. Finally, we note that the interval J = [ OX 0, 0) of L can be written as the union [0 X of intervals of X 0 , liX O]u [ L. Of 2 2X 0] ... U [i< O, i+l X 0] J... There is an order-preserving correspondence of this union with the union [0,1]U [1,2]U ... [i, i+l]d ... of intervals of IR. The latter union equals type of [0,1). When we adjoin the point P with the order type of [0,1]. ] [0,+oa), which has the order O to J, we obtain a set L C.2 Let Definition. L - 9 0 w 0} of be the subspace L' L; it is called the Long Line. every The long line is a path-connected linear continuum, Theorem C.2. point of which has a neighborhood homeomorphic to an open interval of [R. It is not metrizable. Proof. C of ( w of R. Then x < (x0. to that S L be a point of x Let with x lies in the open interval of L, which has the order type of the open interval qT-e fact that L' i& a linear continuum follows from Ex. 6 of §24. x 0) o 0, X (g 0 open intervals 0,AXO) co0 x since they have the point Now let R = / L' ½ in common, of L R countable) subcovering. then S S. in The set If Se. It; L'. [a X To show 0) R with A i>L has no finite (or even is limit point compact, it suffices S in R has a limit point. of first coordinates of points of S And hs an upper is the immediate successor of this upper bound, is a subset of the interval [Gdx O X 0] of Since L'. is a linear continuum, this interval is compact; therefore point. [l We show that the is limit point compact but not compact. to show that every countably infinite set bound in 0 is not compact follows from the fact that the covering by the open sets this is easy: is path connected. L' is not metrizable, so neither is R The fact that of L, each of which is path connected; be the immediate successor of [ [X 0, +) follows that of (0,1) is the union of the L' The result of the preceding paragraph shows that ray Choose an element LOX 0. x S L' has a limit

The Long Line

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