Excluded point topology

Excluded point topology is a specific kind of topology on a set where one specific point is excluded from the open sets of the topology. More formally, let \( X \) be a set and let \( x_0 \in X \) be a designated point. The excluded point topology on \( X \) consists of the following collection of open sets: 1. The empty set \( \emptyset \). 2. The entire set \( X \).