Characterization of Paracompactness in T3 Space/Lemma 4
Jump to navigation
Jump to search
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.
Let $\VV$ be a cover of $T$.
Let $\WW = \set{W \in \tau : \set{V \in \VV : V \cap W \ne \O} \text{ is finite}}$ be an open cover of $T$.
Let $\AA$ be a closed locally finite refinement of $\WW$.
Then:
- $\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite
Proof
Let $A \in \AA$.
By definition of refinement:
- $\exists W \in \WW : A \subseteq W$
From Subsets of Disjoint Sets are Disjoint:
- $\forall V \in \VV : V \cap A \ne \O \leadsto V \cap W \ne \O$
Hence:
- $\set{V \in \VV : V \cap A \ne \O} \subseteq \set{V \in \VV : V \cap W \ne \O}$
We have by hypothesis:
- $\set{V \in \VV : V \cap W \ne \O}$ is finite
From Subset of Finite Set is Finite:
- $\set{V \in \VV : V \cap A \ne \O}$ is finite
Since $A$ was arbitrary, it follows that:
- $\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite
$\blacksquare$