154 Topological Spaces
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
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
Definition (Neighborhood) A property of a subset \(N\) of a topological space \(X\) with respect to a point \(p \in X\), where the following condition applies:
- There exists an open set \(U\) such that \(p \in U \subseteq N\).
where
- \(X\) is a topological space.
- \(p\) is a point in \(X\).
- \(N\) is a subset of \(X\).
- \(U\) is an open subset of \(X\).
154.1 Elementary Example
154.1.1 Simple
A topological space is a set with a topology. Here three points and four open sets.
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ X \} \]
where
- \((X,\tau)\) is the topological space.
154.1.2 General
Hausdorff separation on a discrete four-point space: distinct points have disjoint open neighborhoods.
\[ X = \{ a,\ b,\ c,\ d \} \]
\[ \tau = \mathcal{P}(X) \]
\[ U_{a} = \{a\},\quad U_{b} = \{b\},\quad U_{a} \cap U_{b} = \emptyset \]
where
- \(\mathcal{P}(X)\) is the power set, the discrete topology.
- \(U_{a}, U_{b}\) are disjoint neighborhoods of \(a\) and \(b\).