A super-multiplicative inequality for the number of finite unlabeled arbitrary and $T_0$ topologies

TL;DR

Let $f(n)$ be the number of unlabeled finite topologies on $n$ points and $T(n)$ the number of labeled topologies on $n$ points (with analogous $f_0(n)$ and $T_0(n)$ for $T_0$ topologies). The paper defines a non-commutative $w$-sum operation $\tau_1 \oplus_w \tau_2$ and proves it yields a topology on the union of disjoint ground sets, preserving $T_0$-ness; an injective map from $\mathcal T(n) \times \mathcal T(m)$ to $\mathcal T(n+m)$ is established, yielding the inequalities $f(n+m) \ge f(n) f(m)$ and $T(n+m) \ge T(n) T(m)$ (with analogous bounds for $f_0$ and $T_0$). The main result also provides a max-type bound $f(n+m) \ge \max_{1 \le i \le n+m-1} f(i) f(n+m-i)$. The work clarifies why direct sums fail in the unlabeled case, relates to known counts for small $n$, notes the poset interpretation for $T_0$ topologies, and offers a framework for recursive growth-rate analysis and potential refinements.

Abstract

Let $n$ be a nonnegative integer, and $f(n)$ the number of unlabeled finite topologies on $n$ points. We prove that $f(n+m) \geq f(n) f(m)$ both for the labeled and unlabeled cases. Moreover, we prove a similar inequality for labeled and unlabeled $T_0$ topologies.

Theorems & Definitions (25)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Definition 2.7
• Lemma 2.8
• proof