On the cardinality of the \(T_0\)-topologies on a finite set. (English) Zbl 1408.05007
Summary: Let \(T_0(n,k)\) be the number of all labeled \(T_0\)-topologies having \(k\) open sets that we can define on \(n\) points, and let \(t_0(n,k)\) be the number of those which are nonhomeomorphic. In this paper, we compute these numbers for \(k\geq 5\cdot 2^{n-4}\) and arbitrary \(n\geq 4\). The numbers \(t_{n0}(n,k)\) of all unlabeled and non-\(T_0\)-topologies with \(k\) open sets are also given for \(k\geq 2^{n-2}\).
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
Keywords:
labeled \(T_0\)-topologiesOnline Encyclopedia of Integer Sequences:
Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.Number of topologies, or transitive digraphs with n unlabeled nodes.
Number of topologies on n labeled points satisfying axioms T_0-T_4.
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