A set $\mathbb{E} \in \mathbb{S}$ is open if every point in $\mathbb{E}$ is an interior point.
A set $\mathbb{E}$ is open iff it does not contain any of its boundary points.
For example:
1. open interval - $(2,5), (3,\infty)$.
Unions of open intervals are also open.
2. in $\mathbb{R}^2$ - $\{(x,y) | 12 < x < 2, 0 < y < 1\} $
$\mathbb{R}$ is open set because every point in $\mathbb{R}$ is an interior point.
Think about: what's the meanning of open set?
Closed set is a set that includes all its limit points.
Basically, a close set contain all of its boundary points.
For example:
1. in $\mathbb{R}$ - {p} is closed (single point).
2. (3,6) is an open set, while [3,6] is closed set.
Think about: why we need closed set?
A set could be neither open or closed set, such as (4,7].
$\mathbb{R}$ is closed set because it includes all its limit points.
Closed set is not the opposite of open set, for example, $\mathbb{R}$ is an open and closed set. And How about the empty set?