Defining a subspace

Okay, let's say you have a 4-dimensional space, and you're interested in the subset of points of the form (x,y,x,y), that is, the 1st and third coordinate are always the same, and the 2nd and fourth are always the same. Would this cut a two-dimensional planar subspace in the space because only two values, x and y, are needed to locate any point in the space? Is this the right way to think about this?

Disclaimer: not a mathematician