Abstract
A sequence (z 0,z 1,z 2,, ...,z n, z n+1) of points fromp=z 0 toq=z n+1 in a metric spaceX is said to besequentially equidistant ifd(z i−1,z i)=d(z i,z i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results.
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