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Sierpinski metric

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2010 Mathematics Subject Classification: Primary: 54E35 [MSN][ZBL]

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, 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
How to Cite This Entry:
Sierpinski metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sierpinski_metric&oldid=51440