Covers of Sets in a Topological Space

Covers of Sets in a Topological Space

 Definition: Let $X$ be a topological space and let $A \subseteq X$. Then a Cover/Covering of $A$ is a collection of subsets $\mathcal F$ of $X$ such that $\displaystyle{A \subseteq \bigcup_{U \in \mathcal F} U}$. If $\mathcal F$ is a collection of open sets that satisfies the above inclusion then $\mathcal F$ is called an Open Cover/Covering of $A$. Similarly, if $\mathcal F$ is a collection of closed sets that satisfies the above inclusion then $\mathcal F$ is called a Closed Cover/Covering of $A$. If $\mathcal F^* \subseteq \mathcal F$ also covers $A$ then we said that $\mathcal F^*$ is a Subcover of $A$ from $\mathcal F$.

Sometimes the term “cover” is used to denote an “open cover” when the context is clear and no ambiguity can arise.

For example, consider the topological space $\mathbb{R}$ with the usual topology. Consider the set $A = \mathbb{N} \subset \mathbb{R}$. Every singleton set in $\mathbb{R}$ is closed and so the collection $\{ \{ n \} : n \in \mathbb{N} \}$ is a closed covering of $\mathbb{N}$ since:

(1)
\begin{align} \quad \mathbb{N} = \bigcup_{n \in \mathbb{N}} \{ n \} \end{align}

On the otherhand, the collection $\{ \left ( n - \frac{1}{2}, n + \frac{1}{2} \right ) : n \in \mathbb{N} \}$ is an open covering of $\mathbb{N}$ as every set in this collection is an open interval centered at $n$ with radius $\frac{1}{2}$, and:

(2)
\begin{align} \quad \mathbb{N} \subset \bigcup_{n \in \mathbb{N}} \left ( n - \frac{1}{2}, n + \frac{1}{2} \right ) \end{align}

For another example, consider the set $A = [0, 1) \subset \mathbb{R}$. Then the collection $\mathcal F = \left \{ \left (-1, \frac{2}{3} \right ), \left ( \frac{1}{2}, 2 \right ) \right \}$ is an open cover of $[0, 1)$.

In general, many open covers for subsets of a topological space may exist.