Let , let be the subspace
Prove is a closed map but not an open map.
The space X is just the union of the two coordinate axes. The mapping f (projection onto the first coordinate) acts as the identity on the x-axis, and takes everything on the y-axis to the origin. You should be able to see from that that the image under f of a closed set will always be closed (think of the set as the union of two parts, the part on the x-axis and the part on the y-axis: the image under f of each of these two parts is fairly obviously closed).
The map is not open, because for example the image of the open subset of X is a single point and therefore not open.