Open Set

Topics: Calculus - Topology

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(definition)

Let $A\subseteq {\mathbb{R}}^n$. We say that $A$ is an open set if and only if $A=\text{int}(A)$. All of the points in A are interior.

Open does not imply not Closed

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

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

Union and Intersection §

(theorem)

Let $A$ and $B$ be two open sets, then $A\cup B$ is open and $A\cap B$ is also open (union and intersection).

From this theorem, it also follows that if $A$ and $B$ are closed, then their union and intersection are also closed.