Metric characterization and the integers (Q1085830)

In the definition of te well-known concept of a point of order n (in the sense of Menger-Urysohn) the author replaces an arbitrary neighborhood of a point with a metric ball, and he obtains the following notion. A point x in a metric space (X,d) is said to be of metric order n if for each \(r>0\) we have card\(\{\) \(y\in X:\) \(d(x,y)=r\}=n\). A space (X,d) is said to have metric order n if all its points have metric order n. It is proved that if (X,d) is locally compact and of metric order n, then n cannot be odd; and in case \(n=2\) the real line \(E^ 1\) and the product \(S^ 1\times \omega\) of the circle \(S^ 1\) by the nonnegative integers \(\omega\) are the only locally connected examples. Some related results are also discussed.