Neighborhood in Open Subspace
Jump to navigation
Jump to search
Theorem
Let $T = \struct{S, \tau}$ be a topological space.
Let $U \subseteq S$ be an open subset.
Let $\tau_U$ denote the subspace topology on $U$.
Let $s \in U$
Let $N \subseteq U$ be a subset.
Then:
- $N$ is a neighborhood of $s$ in $\struct{U, \tau_U}$
- $N$ is a neighborhood of $s$ in $\struct{S, \tau}$
Proof
Necessary Condition
Let $N$ be a neighborhood of $s$ in $\struct{U, \tau_U}$.
By definition of neighborhood:
- $\exists V \in \tau_U : x \in V \subseteq N$
From Open Set in Open Subspace:
- $V \in \tau$
Hence:
- $\exists V \in \tau : x \in V \subseteq N$
It follows that $N$ is a neighborhood of $s$ in $\struct{S, \tau}$ by definition.
$\Box$
Sufficient Condition
Let $N$ be a neighborhood of $s$ in $\struct{S, \tau}$.
By definition of neighborhood:
- $\exists W \in \tau : x \in W \subseteq N$
Let $V = W \cap U$.
From Subset Relation is Transitive:
- $V \subseteq N$
By definition of topological space:
- $V \in \tau$.
From Open Set in Open Subspace:
- $V \in \tau_U$
Hence:
- $\exists V \in \tau_U : x \in V \subseteq N$
It follows that $N$ is a neighborhood of $s$ in $\struct{U, \tau_U}$ by definition.
$\blacksquare$