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 $\tau$ on X is a set of subsets $U\subset X$ (called open sets) such that

1. $\varnothing$, $X\in \tau$
2. Closed under arbitrary union: for an index set I (not necessarily countable or finite - can be arbitrary) for $U_{i\in I}\in \tau$ $$\bigcup _{i\in I}{U}_i\in \tau$$
3. Closed under finite intersection: for $n\in \mathbb{N}$, $U_i\in \tau$ for $i=1,...,n$ $$\bigcap _{i=1}^n{U}_i\in \tau$$

Common topologies

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