Subspace topology

Definition
Let $$(X,\tau)$$ be a topological space (viz, a set $$X$$ endowed with a topology $$\tau$$) and $$A$$ be a subset of $$X$$. The subspace topology or induced topology or relative topology on $$A$$ can be defined in many equivalent ways. Note that $$A$$ induced with this topology is a topological space in its own right. Thus, subsets of topological spaces are often also called subspaces.

The subspace topology can be defined in many equivalent ways. The equivalent formulations are described below:

Textbook references

 * , Page 88-89
 * , Page 10 (Theorem 6): introduced under the name relative topology