Definition 3.1.10 (Subspace Topology).label Let $(X,\tau_{X})$ be a topological space and $A\suf X$ be a subset then the subspace topology on $A$
\begin{align*}\tau_{A}=\curl{U\cap A:U\in\tau_X}\end{align*}
is the initial topology on $A$ with respect to the inclusion map $\iota:A\inj X$.
Proof. Observe
\begin{align*}\iota\inv(U)=\curl{a\in A:\iota(a)\in U}=\curl{a\in A:a\in U}=U\cap A\end{align*}
for any subset $U\suf X$ hence the same holds for $U\in\tau_{X}$.$\square$
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