Closed Set

Topics: Calculus - Topology

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(definition)

Let $A\in {\mathbb{R}}^n$. We say that $A$ is closed if and only if $A^c$ (its complement) is open.

Closed does not imply not Open

Remember that a set being closed doesn’t imply it’s not open. Similarly, a set being open doesn’t imply it’s not closed.

The empty set is an example of a set that is closed and open at the same time.

Example 1

Let $A\subseteq \mathbb{R}$, $A=[0,1]$.

We have that $A^c=(-\infty ,0)\cup (1,\infty )$. Notice that $A^c$ is open, so it follows that $A$ is closed.

Example 2

Let $B=[2,3)$.

We have that $B^c=(\infty ,2)\cup [3,\infty )$. Notice that $B^c$ isn’t open, so it follows that $B$ isn’t closed.

Notice that $B$ also isn’t open (but this doesn’t follow the closeness of $B^c$).

Example 3

Let $C=(-\infty ,0)$.

We have that $C^c=[0,\infty )$. Notice that $C^c$ isn’t open, so it follows that $C$ isn’t closed. However, do notice that $C$ is open.

Union and Intersection §

(corollary, from this theorem)

If $A$ and $B$ are closed, then $A\cup B$ and $A\cap B$ are also closed.

Using the Limit Points of a Sequence §

(lemma, from the definition of a limit point)

Let $A\subseteq {\mathbb{R}}^n$.

$A$ is closed if $\forall \text{ convergent }\{a_k\}\subseteq A$ (a sequence), the limit point of $a_k$ is in $A$.