A subspace Y of a topological space X is just a subset of X. However, one needs to impose a topology on Y to make it a subspace.
Definition 4.9.1 (Subspace topology) Let X be a topological space, and Y be a subset of X. The subspace topology on Y has as opens the subsets of Y of the form U ∩ Y, U open in X.
One then says that Y, with the subspace topology, is a subspace of X.
There is a slightly more general definition.
Definition 4.9.2 (Induced topology) Given any map f : Z → X, where Z is a set and X is a topological space, the topology on Z induced from that of X by f is the coarsest topology on Z that makes f continuous.
The induced topology is simple: take as opens in Z exactly those subsets of the form f−1(U), U open in