On the Structure of Finite T0 + T5 Spaces

Abstract

The object of this paper is to study some structural aspects of finite T0 + T4 and T0 + T5 spaces in order to establish certain recursion relations that can be used to obtain the number of (labelled as well as unlabelled) T0 + T5 topologies on a finite set.Here, as in [2], a topology 𝒥 is a T4(T5) space provided for any pair of disjoint closed sets A and B (separated sets A and B = A ∩ closure B = B ∩ closure A = 0) there exist disjoint open sets 0A and 0B 𝒥 such that A ⊆ 0A and B ⊆ 0B. An almost immediate consequence of these investigations is that the inherent simplicity of the connected T0 + T5 topologies ensures that they are reconstructable.