Ultrafilter on a set

An **ultrafilter** on a set \( X \) is a special type of filter that has additional properties, particularly in topology and set theory. Here's a precise definition and some key properties: 1. **Filter**: A filter \( \mathcal{F} \) on a set \( X \) is a collection of subsets of \( X \) such that: - The empty set is not in \( \mathcal{F} \).