An open set in topology is a set that contains all of its interior points. Interior points are those points that are not boundary points of the set.

The complement of an open set is a closed set. A closed set is a set that contains all of its limit points.

A point $x$ is called a limit point of a set $S$ if every open neighborhood of $x$ contains a point of $S$ other than $x$.

A neighborhood of a point $x$ is a set that contains $x$.

A function $f$ is continuous at a point $x$ if the limit of $f(x)$ as $x$ approaches $a$ is equal to $f(a)$.

A function $f$ is continuous on an interval $I$ if it is continuous at every point of $I$.

The set of all real numbers between 0 and 1 is open because it does not contain any of its boundary points.

The set of all integers is neither open nor closed because it contains some of its boundary points but not all of them.

The set of all rational numbers is neither open nor closed because it contains some of its boundary points but not all of them.

The set of all irrational numbers is neither open nor closed because it contains some of its boundary points but not all of them.

The set of all points in the plane that are equidistant from the origin is called a circle.

The set of all points in the plane that are on the same line as the origin is called a line.

The set of all points in the plane that are not on the same line as the origin is called a plane.

The set of all points in the plane that are in the first quadrant is called a quadrant.

The set of all points in the plane that are in the first octant is called an octant.