139 Hausdorff Space
A Hausdorff space is a topological space in which any two distinct points have disjoint neighborhoods that is used to guarantee uniqueness of limits.
Definition (Hausdorff Space) A Hausdorff space is a topological space, \(X\), that satisfies the following condition:
- Disjoint neighborhoods: \(U_1 \cap U_2 = \empty\), for any pair of points \(p_1,p_2 \in X\)
where
- \(U_1, U_2\) are neighborhoods
- \(p_1,p_2 \in X\) are distinct points
- \(X\) is a topological space
139.1 Elementary Example
139.1.1 Simple
In a Hausdorff space, distinct points have disjoint neighborhoods.
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \mathcal{P}(X) \]
\[ p_{1} = 1,\ p_{2} = 2,\quad U_{1} = \{1\},\ U_{2} = \{2\} \]
\[ U_{1} \cap U_{2} = \emptyset \]
where
- \(U_{1}, U_{2}\) are neighborhoods of \(p_{1}, p_{2}\).
- \(\mathcal{P}(X)\) is the discrete topology.