Coverings of a Subset in Euclidean Space

# Coverings of a Subset in Euclidean Space

 Definition: Let $S \subseteq \mathbb{R}^n$. A Covering of $S$ is a collection of subsets of $\mathbb{R}^n$, $\mathcal F$, such that $\displaystyle{S \subseteq \bigcup_{A \in \mathcal F} A}$. If $\mathcal F$ is a collection of open sets, then $\mathcal F$ is called an Open Covering of $S$ and if $\mathcal F$ is a collection of closed sets, then $\mathcal F$ is called a Closed Covering of $S$.

For example, consider the subset $S = (0, 1) \subseteq \mathbb{R}^n$ and the collection of sets:

(1)
\begin{align} \quad \mathcal F = \left \{ \left (0, 1 - \frac{1}{n} \right ) : n \in \mathbb{N} \right \} = \left \{ (0, 0), \left (0, \frac{1}{2} \right ), \left ( 0, \frac{2}{3} \right ) , ..., \left ( 0, 1 - \frac{1}{n} \right ), ... \right \} \end{align}

It's not hard to see that $\mathcal F$ is a covering of $S = (0, 1)$. Furthermore, the collection $\mathcal F^* = \left \{ \left (0, \frac{1}{2} \right ), \left [ \frac{1}{2}, 1 \right ) \right \}$ is also a covering of $S = (0, 1)$. In particular, $\mathcal F^*$ is a finite covering of $S$.

For another example, consider the subset $S = \{ (x, y) \in \mathbb{R}^2 : x \geq 0 , y \geq 0 \}$. Furthermore consider the following collection of sets:

(2)
\begin{align} \quad \mathcal F = \left \{ B(\mathbf{0}, n) : n \in \mathbb{Z} \right \} = \{ B(\mathbf{0}, 1), B(\mathbf{0}, 2), ..., B(\mathbf{0}, n), ... \} \end{align}

Once again, it should not be too hard to see that $\mathcal F$ is a covering of $S$: We see that this is a countably infinite covering of $S$.