View source for Sierpinski metric

{{TEX|done}}{{MSC|54E35}}

A [[metric]] on a countably infinite set $X = \{x_1,x_2,\ldots\}$.  For $i \ne j$ define $d(x_i,x_j) = 1 + 1/(i+j)$, and $d(x_i,x_i) = 0$.  The Sierpinski metric is [[Complete metric space|complete]], since every [[Cauchy sequence]] is ultimately constant.  The induced topology is the [[discrete topology]].

====References====
* Steen, Lynn Arthur; Seebach, J.Arthur jun. ''Counterexamples in topology'' (2nd ed.) Springer (1978) {{ISBN|0-387-90312-7}} {{ZBL|0386.54001}}