Proposition Let \((X,d)\) be a metric space and \(Y \subseteq X\). Then \(d \restriction Y^2\) is a metric on \(Y\) that induces the subspace topology. Proposition Every metric space is Hausdorff. Proposition Let \(((X_n, d_n))\) be a sequence of metric spaces. Define \(D\) on \(prod_n X_n\) by \(D((x_n)) = \sup_n (\overline{d_n}(x_n, y_n)/n)\), where \(\overline{d_n}\) is the standard bounded metric corresponding to \(d_n\). Then \(D\) is a metric on \(\prod_n X_n\) that induces the product topology. Theorem 21.1 Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. Let \(f : X \rightarrow Y\). Then \(f\) is continuous if and only if, for all \(x \in X\) and \(\epsilon > 0\), there exists \(\delta > 0\) such that, for all \(y \in X\), \[ d_X(x,y) Definition Let \(X\) be a topological space. Let \(x \in X\). A basis at the point \(x\) is a set \(\mathcal{B}\) of neighbourhoods of \(x\) such that every neighbourhood of \(x\) includes a member of \(\mathcal{B}\)...
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