Put simply, a subspace is analogous to a subset of a topological space. Subspaces have powerful applications in topology.
Let be a topological space, and let be a subset of . Define the open sets as follows:
A set is open in if there exists a a set such that
An important idea to note from the above definitions is that a set not being open or closed does not prevent it from being open or closed within a subspace. For example, as a subspace of itself is both open and closed.