PreprintCC BY 4.0arXiv
On the Number of Distinct Topological Bases of a Finite Set of Size N
Abstract
For a finite set S with lvert Srvert = N, the number of families ℬ ⊆ 𝒫(S) that are topological bases is #(N) = ∑𝒯 ∈ Top(S) 2lvert𝒯rvert − lvertℳ𝒯rvert, where ℳ𝒯 is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating 𝒯 are exactly the sets with ℳ𝒯 ⊆ ℬ ⊆ 𝒯. The small-N table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.
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BibTeX
@misc{ericson2026number,
title = {{On the Number of Distinct Topological Bases of a Finite Set of Size $N$}},
author = {Lars Warren Ericson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14102},
eprint = {2609.14102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14102},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Number of Distinct Topological Bases of a Finite Set of Size N
AU - Lars Warren Ericson
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14102
UR - https://arxiv.org/abs/2609.14102
AB - For a finite set S with lvert Srvert = N, the number of families ℬ ⊆ 𝒫(S) that are topological bases is #(N) = ∑_𝒯 ∈ Top(S) 2^lvert𝒯rvert − lvertℳ_𝒯rvert, where ℳ_𝒯 is the canonical minimal basis of minimal open neighborhoods. The identity is proved in Lean 4 / Mathlib (`CARDB.lean`): bases generating 𝒯 are exactly the sets with ℳ_𝒯 ⊆ ℬ ⊆ 𝒯. The small-N table and the discrete-dominance sandwich are proved in `CARDB/SmallN.lean` and `CARDB/Asymptotics.lean`.
ER -