145 Neighborhood
A neighborhood is an open set containing a given point that is used to describe local behavior of a function near that point.
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 \(X\).
145.1 Elementary Example
145.1.1 Simple
A neighborhood of a point contains an open set that contains that point.
\[ X = \{ 1,\ 2,\ 3 \} \]
\[ \tau = \{ \emptyset,\ \{1\},\ \{1,2\},\ X \} \]
\[ p = 1,\quad U = \{1\},\quad N = \{1,2\} \]
\[ p \in U \subseteq N,\quad U \in \tau \]
where
- \(p\) is the point.
- \(U\) is an open set.
- \(N\) is a neighborhood of \(p\).