In topology and related fields of mathematics, a set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U. In other words, the distance between any point x in U and the edge of U is always greater than zero.
As an example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. Here, the topology is the usual topology on the real line. We can look at this in two ways. Since any point in the interval is different from 0 and 1, the distance from that point to the edge is always non-zero. Or equivalently, for any point in the interval we can move by a small enough amount in any direction without touching the edge and still be inside the set. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if one takes x = 1 and moves even the tiniest bit in the positive direction, one will be outside of (0,1].
Reference:
http://en.wikipedia.org/wiki/Open_set
