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

TL;DR

This paper establishes a super-multiplicative inequality for the number of finite unlabeled and labeled topologies, including the special case of $T_0$ topologies, revealing new combinatorial properties.

Contribution

It proves a super-multiplicative inequality for the counts of finite topologies and $T_0$ topologies, extending known results to unlabeled cases.

Findings

Proves $f(n+m) \\geq f(n)f(m)$ for unlabeled topologies.

Establishes similar inequality for labeled topologies.

Extends results to $T_0$ topologies.

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.