Subset of Cover is Cover of Subset

Theorem

Let $S$ be a set.

Let $\CC$ be a cover of $S$.

Let $T \subseteq S$.

Let:

$\CC_T = \set {C \in \CC : C \cap T \ne \O}$

Then $\CC_T$ is a cover of $T$:

$T \subseteq \bigcup \CC_T$

Proof

Let $x \in T$.

By definition of a cover:

$\exists C \in \CC : x \in C$

By definition of set intersection:

$x \in C \cap T$

Hence:

$C \in \CC_T$

Since $x$ was arbitrary, it follows that $\CC_T$ is a cover of $T$ by definition and:

$T \subseteq \bigcup \CC_T$

$\blacksquare$