Properties of topological spaces

Topological spaces are fundamental objects of study in topology, a branch of mathematics focused on the properties of space that are preserved under continuous transformations. Here are some key properties and concepts associated with topological spaces: 1. **Open and Closed Sets**: - A topology on a set \(X\) is a collection of subsets of \(X\) (called open sets) that includes the empty set and \(X\) itself, and is closed under arbitrary unions and finite intersections.