Closure

The closure of a subset $E \sube X$, denoted $\bar{E}$, contains $E$ and its limit points.

$$ \bar{E} = E \cup E' $$

Moreover, we call the set of all interior points its interior denoted $E^o$.

Closure is Smallest Closed Set Containing $E$. The closure of $E \sube X$ is the intersection of all closed subsets containing $E$.

$$ \bar{E} = \bigcap _{A \in \mathcal{S}} A \text{ with } \mathcal{S} = \{A \sube X \space | \space A \text{ is closed and } E \sube A \} $$

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Closure is Closed. If $E \sube X$, then $\bar{E}$ is closed in $X$.

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Equal to Closure iff Closed. $E \sube X$ is closed in $X$ iff $E = \bar{E}$.

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Closed Ambient Set contains Closure. If $A$ is a closed subset relative to a metric space $X$ and $E \sube A$, then $\bar{E} \sube A$.

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Open iff Intersected by Open Superset. For $E \sube X \sube Y$, $E$ is open in $X$ iff there is an open superset for which $E$ is the intersection of $X$ with it.

$$ \text{$E$ is open} \iff \exist U \sube Y, U\text{ is open} : E = U \cap X $$

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Open Cover

An open cover of a subset $E \sube X$ is a collection of open subsets $\{ U \alpha \}$ such that $E \sube \bigcup{\alpha} U_\alpha$.

Compactness

A subset $K \sube X$ is compact if every open cover of $K$ has a finite subcover. That is, given an open cover $\{ U_\alpha\}$ of $K$, there is a finite subcollection $\{ U_{\alpha _i} \} {i=1} ^n$ such that $K \sube \{ U{\alpha _i} \} _{i=1} ^n$.

Any open cover has some redundancy.