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The Number of Topologies on a Finite Set
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Moussa Benoumhani

Faculty of Science

Department of Mathematics

Sana'a University

P.O. Box 14026

Sana'a

Yemen

**Abstract:**
Let *X* be a finite set having *n* elements. How many different
labeled topologies one can define on *X*? Let *T(n,k)* be
the number of topologies having *k* open sets. We compute
*T(n,k)* for 2 <= *k* <= 12, as well as other results
concerning *T*_{0} topologies on *X*
having *n*+4 <= *k* <= *n*+6
open sets.

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(Concerned with sequences
A000798
A001930 and
A008277
.)

Received August 29 2005;
revised version received May 20 2006.
Published in *Journal of Integer Sequences* May 24 2006.

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