Show that if \(Y\) is a subspace of \(X\), and \(A\) is a subset of \(Y\), then the topology \(A\) inherits as a subspace of \(Y\) is the same as the topology it inherits as a subspace of \(X\).
Solution
Let \(\mathcal T_X\) be the topology of \(X\). The topology that \(A\) inherits directly from \(X\) is \[ \{A\cap U:U\in\mathcal T_X\}. \] The topology that \(Y\) inherits from \(X\) is \(\{Y\cap U:U\in\mathcal T_X\}\). Hence the topology that \(A\) inherits from \(Y\) consists of \[ A\cap(Y\cap U)=(A\cap Y)\cap U=A\cap U \] with \(U\in\mathcal T_X\), since \(A\subseteq Y\). Thus the two subspace topologies coincide.