Topological Spaces

Topological spaces are abstract mathematical structures defined by a set and a collection of open sets, forming the foundation of topology.

Definition

A topological space is a pair \((X, \tau)\), where \(X\) is a set and \(\tau\) is a topology: a collection of subsets satisfying open set axioms.

Examples

\(\mathbb{R}\) with open intervals as \(\tau\) (standard topology) or \(\{\emptyset, \mathbb{R}\}\) (trivial topology).

Importance

They generalize metric spaces, enabling the study of continuity and convergence without distances.