Subspace topology

In general topology, the subspace topology, or induced or relative topology, is the assignment of open sets to a subset of a topological space.

Let (X,T) be a topological space with T the family of open sets, and let A be a subset of X. The subspace topology on A is the family

${\mathcal {T}}_{A}=\{A\cap U:U\in {\mathcal {T}}\}.\,$

The subspace topology makes the inclusion map A → X continuous and is the coarsest topology with that property.

References

• Wolfgang Franz (1967). General Topology. Harrap, 36. 
• J.L. Kelley (1955). General topology. van Nostrand, 50-53.