The unit square in a Cartesian coordinate system with coordinates (x,y) is defined as the square consisting of the points where both x and y lie in the unit interval from 0 to 1.
Whether the interval used is open or closed, and at which ends, is at the discretion of the user of the term, who should make it clear which precise definition they are using.
However, the term unit interval is most usually used to refer to the closed interval [0,1].
* unit sphere
* unit cube
* Eric W. Weisstein, Unit square at MathWorld.