Open set

A set where every point has some wiggle room -- a small enough ball around it still fits inside.

A set $U\subseteq\mathbb{R}^n$ is open if every point $\mathbf{a}\in U$ has some radius $\varepsilon>0$ for which the whole ball $B_\varepsilon(\mathbf{a})$ still lies in $U$ . No point of an open set sits exactly on its edge -- every point has breathing room in every direction.

Examples

The open disk $\{(x,y): x^2+y^2<1\}$ is open -- every interior point has a small ball around it still inside the disk.
The half-plane $\{(x,y): x>0\}$ is open, but $\{(x,y): x\geq0\}$ is not (points with $x=0$ have no wiggle room to the left).