153 Topological Space
A topological space is a set equipped with a topology that is used to study continuity of mappings without requiring a metric.
Definition (Topological Space) A topological space is a pair \((X, \mathcal{T})\) on \(X\), where
- \(X\) is a set
- \(\mathcal{T}\) is a topology
153.1 Elementary Example
153.1.1 Simple
A topological space is a pair \((X,\tau)\) with \(\tau\) a topology on \(X\).
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ X \} \]
\[ (X,\tau) \]
where
- \(X\) is the set.
- \(\tau\) is the topology.
- \((X,\tau)\) is the topological space.