22 - Covers

Open Cover

Let $X$ be a metric space with $\mathcal{U}=\{U_i:i\in I\}$ a collection of open subsets of $X$

$\mathcal{U}$ is an open cover of $X$ if

$$X=\bigcup _{i\in I}{U}_i$$

Note that for subspaces $Y\subseteq X$ then need $Y\subseteq {\bigcup }_{i\in I}{U}_i$

Subcover

Let $X$ be a metric space with open cover and let

$$\mathcal{U}=\{U_i:i\in I\}\text{ be an open cover of }X$$

If $J\subseteq I$ then
sub-collection $\{U_i:i\in J\}$ is a subcover of $\mathcal{U}$ if

$$X=\bigcup _{i\in J}{U}_i$$

If $|J|<\infty$ then $\{U_i:i\in J\}$ is a finite subcover

Note that for subspaces $Y\subseteq X$ then need $Y\subseteq {\bigcup }_{i\in J}{U}_i$