Integer programming is a tool for solving some combinatorial optimization problems. In thispaper, we deal with combinatorial optimization problems onfinite topologies. We use the binaryrepresentation of the sets to characterize finite topologies as the solutions of a Boolean quadraticsystem. This system is used as a basic model for formulating other types of topologies (e.g. doortopology andT0-topology) and some combinatorial optimization problems on finite topologies. Asan example of the proposed model, we found that the smallest numberm(k)for which the topologyexists on anm(k)-elements set containing exactlykopen sets, fork=8 andk=15 is 3 and 5,respectively.

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