Characterization of Paracompactness in T3 Space/Lemma 19
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 $\BB$ be a discrete set of subsets of $X$.
Let $\UU = \set{ U \in \tau : \size {\set{B \in \BB : U \cap B} } \le 1}$
Then:
- $\UU$ is a open cover of $X$ in $T$.
Proof
Let $s \in X$.
By definition of discrete:
- $\exists U \in \tau : x \in U : \size {\set{B \in \BB : U \cap B} } \le 1$
Hence:
- $U \in \UU$
Since $x$ was arbitrary:
- $\forall x \in X : \exists U \in \UU : x \in U$
It follows that $\UU$ is an open cover of $X$ in $T$ by definition.
$\blacksquare$