147 Open Sets
An open set is a member of a topology on a set that is used to define neighborhoods and continuity of mappings.
The open set is central to topology, and it is required for definitions. We begin with a definition and then an example.
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 \(X\)
- \(U \subseteq X\)
147.1 Elementary Example
147.1.1 Simple
Open sets are the members of a topology. On \(X = \{1,2,3\}\), the set \(\{1\}\) is open in the listed topology.
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ X \} \]
\[ \{1\} \in \tau \]
where
- \(\tau\) is the topology.
- \(\{1\}\) is an open set.
147.1.2 General
Unions of open sets are open. On four points, unite two open sets.
\[ X = \{ 1,\ 2,\ 3,\ 4 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{2\},\ \{1,2\},\ X \} \]
\[ \{1\} \cup \{2\} = \{1,2\} \in \tau \]
where
- \(\{1\}\) and \(\{2\}\) are open.
- their union is open.
Example 1 (Open Set) Let
\[ X = \{1,2,3\} \]
\[ \mathcal{T} = \{ \emptyset, \{1\}, \{1,2\}, X \}. \]
Then, \((X,\mathcal{T})\) is a topological space, where
*\[ (X,\mathcal{T}) = \left( \{1,2,3\}, \{ \emptyset, \{1\}, \{1,2\}, \{1,2,3\} \} \right) \]
Let
\[ U = \{1,2\}. \]
Since
\[ U \in \mathcal{T}, \]
it follows that \(U\) is an open set.
Therefore,
\[ \{1,2\} \]
is an open set in \((X,\mathcal{T})\).