T4 Property Preserved in Closed Subspace/Corollary
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a topological space.
Let $T_K$ be a subspace of $T$ such that $K$ is closed in $T$.
If $T$ is a normal space then $T_K$ is also a normal space.
That is, the property of being a normal space is weakly hereditary.
Proof
From the definition, $T = \struct {S, \tau}$ is a normal space if and only if:
- $\struct {S, \tau}$ is a $T_4$ space
- $\struct {S, \tau}$ is a $T_1$ (Fréchet) space.
From Separation Properties Preserved in Subspace, any subspace of a $T_1$ space is also a $T_1$ space.
From T4 Property Preserved in Closed Subspace, any closed subspace of a $T_4$ space is also a $T_4$ space.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces