By Eisenhart L.P.

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A, b E Ij and let f nO(ab)be the arc with the longest length in { f "(ab) : n E Z). Let E > 0. Then we can choose Ij and a, b E Ij such that the length between h fno(a) and fnO(b) is less than E . Then, diam(fn(ab)) < E for all n E Z and therefore f : S1 -+ S1 is not expansive. This is a contradiction. The following theorem was proved in Hiraide [Hi51 and Lewowicz [L2]. 27. S2. There exists no expansive homeomorphism of the bsphere We omit the proof. If the theorem is of interest, the reader should see [Hi51 and [L2].

If any two points in X are joined by a path, then X is said to be path connected. In general, a connected space need not be path connected. An arc in X is an injective continuous map from [ O , l ] to X. We say that X is arcwise connected if any two points in X are joined by an arc. It is clear that if X is arcwise connected then it is path connected. 3. A Hausdorfl space X is path connected if and only if X is arcwise connected. CHAPTER 2 34 Proof. Let f : [O,1] -t X be any path with f (0) # f (1).

4. If a compact metric space X is locally connected, then X is locally path connected. Moreover every connected component of X is path connected. Proof. Let E > 0. Since X is locally connected, there exists a sequence of numbers 6, > 0 such that any two points with d ( x ,X I ) < 6, are contained in a connected set of diameter less than €12,. We show that any two points x(0) and x(1) with d(x(O),x(l))< 60 can be . do this choose a sequence of jointed by a path of diameter at most 4 ~ To , eack of which divides the next.