User:Leigh.Samphier/Topology/T3 Space is Fully T4 iff Paracompact
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Theorem
Let $T = \struct {S, \tau}$ be a $T_3$ space.
Then $T$ is fully $T_4$ if and only if $T$ is paracompact.
Proof
Necessary Condition
$\Box$
Sufficient Condition
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Compactness Properties and the $T_i$ Axioms