In
mathematics in general, a subspace is a subset of another space that is
closed in regard to the
operations that are being used and/or retains the key properties.
The most common example: a subspace S of a vector space V is a subset of V that is closed in regard to vector addition, i.e. the sum of any two elements of S is again an element of S. For euclidean space, this is easy to visualize: a subspace of three-dimensional space is simply a plane or a straight line that goes through the origin.
Another, rather more obscure example: a subspace (S, OS) of a topological space (X, O) is composed of a subset S of X and a subset OS of O that is composed of all the intersections between S and the elements of O.