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.