146 Open Set
An open set is a member of a topology on a set that is used to define neighborhoods and continuity of mappings.
Definition (Open Set) A subset \(U\) of a topological space, \((X,\mathcal{T})\), where the following condition applies:
- \(U \in \mathcal{T}\)
where
- \(X\) is a set
- \(\mathcal{T}\) is a topology on \(X\)
- \(U \subseteq X\)
146.1 Elementary Example
146.1.1 Simple
An open set is a member of the topology \(\tau\).
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ X \} \]
\[ U = \{1\} \in \tau \]
where
- \(U\) is an open set.
- \(\tau\) is the topology on \(X\).