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Counting Finite Topologies

Eldar Fischer, Johann A. Makowsky

Abstract

In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.

Counting Finite Topologies

Abstract

In this paper we study the number of finite topologies on an -element set subject to various restrictions.
Paper Structure (15 sections, 11 theorems, 23 equations)

This paper contains 15 sections, 11 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Background
  4. C-finite and MC-finite sequences of integers
  5. Counting finite topologies
  6. Topologies as relational structures
  7. The logics $\mathrm{TCMSOL}$ and $\mathrm{CMSOL}$
  8. Hard-wired constant symbols
  9. Topological model theory
  10. Most general Specker-Blatter Theorem
  11. Proof of Theorem \ref{['th:main']}
  12. Alexandroff's Theorem
  13. Translation schemes
  14. Conclusions
  15. Normal sequences

Key Result

Theorem 1

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • Theorem 7
  • Theorem 8
  • proof
  • Theorem 11: Most general Specker-Blatter Theorem
  • proof
  • Theorem 13: Alexandroff, 1931
  • ...and 6 more