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Subspace Topology**

Given a topological space \(X\) with topology \(\mathcal{T}\), any subset \(A\subseteq X\) can be made into a topological space by the 'subspace topology' \(\mathcal{T}|_A=\{A\cap U\subseteq A|U\in \mathcal{T}\}\), where the open subsets of \(A\) are the intersections of \(A\) with the open subsets of \(X\).