The document discusses a projection map pi_1 from R^2 to R that projects onto the first component. It defines the subspace A of R^2 as the union of points where x is greater than or equal to 0 and points where y is equal to 0. The projection map pi_1 restricts to an open mapping from the subspace A to R because the projection of the first component x maps from -infinity to +infinity, which is an open interval. However, the projection map is not a closed mapping, as shown by considering the image of the tangent function under projection, which is always an open, not closed, interval.