Characterization of Paracompactness in T3 Space/Statement 5 implies Statement 6

This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code.

Theorem

Let $T = \struct{X, \tau}$ be a topological space.

If every open cover of $T$ has an open $\sigma$-discrete refinement then:

every open cover of $T$ has an open $\sigma$-locally finite refinement

Proof

Follows immediately from Sigma-Discrete Set of Subsets is Sigma-Locally Finite.

$\blacksquare$

Sources

• 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces: $\S$Paracompactness: Theorem $28$