Subspace Topology

In the setting of Sets and Maps: Ordinary Notation, let $(X,\mathcal{T})$ be a topological space and let $A$ be a subset of $X$.

(i) The subspace topology on $A$ is

$$\mathcal{T}_{A}=\{W\in\mathcal{P}(A):W=A\cap U\text{ for some }U\in\mathcal{T}\},$$

a topology on $A$ by The Subspace Topology is a Topology (i); thus $(A,\mathcal{T}_{A})$ is a topological space.

(ii) A subset $W$ of $A$ is open in $(A,\mathcal{T}_{A})$ if and only if $W=A\cap U$ for some $U\in\mathcal{T}$.