topology

Background

A generalization of open sets, also called topological space (a set X combined with a topology)

Definition

For a set X, a topology on on X is a set of subsets (called open sets) such that

1. ,
2. Closed under arbitrary union: for an index set I (not necessarily countable or finite - can be arbitrary) for
3. Closed under finite intersection: for , for

Common topologies

• Euclidean topology
• lower limit topology
• discrete topology
• finite complement topology
• subspace topology
• metric topology