Let \(X\) be the subset of \(\mathbb{R}^\omega\) consisting of all sequences \(\mathbf{x}\) such that \(\sum x_i^2\) converges. Then the formula
\[ d(\mathbf{x}, \mathbf{y}) = \left[ \sum_{i=1}^{\infty} (x_i - y_i)^2 \right]^{1/2} \]
defines a metric on \(X\). (See Exercise 10.) On \(X\) we have the three topologies it inherits from the box, uniform, and product topologies on \(\mathbb{R}^\omega\). We have also the topology given by the metric \(d\), which we call the \(\ell^2\)-topology.
- Show that on \(X\), we have the inclusions
\[ \text{box topology} \supset \ell^2\text{-topology} \supset \text{uniform topology}. \]
The set \(\mathbb{R}^\infty\) of all sequences that are eventually zero is contained in \(X\). Show that the four topologies that \(\mathbb{R}^\infty\) inherits as a subspace of \(X\) are all distinct.
The set
\[ H = \prod_{n \in \mathbb{Z}_+} [0, 1/n] \]
is contained in \(X\); it is called the Hilbert cube. Compare the four topologies that \(H\) inherits as a subspace of \(X\).
Solution
Write \(\mathcal T_{\mathrm{box}}\), \(\mathcal T_d\), \(\mathcal T_{\bar\rho}\), and \(\mathcal T_{\mathrm{prod}}\) for the box, \(\ell^2\), uniform, and product topologies on \(X\) and on its subspaces. On \(\mathbb R^\omega\), \(\mathcal T_{\mathrm{prod}}\subseteq\mathcal T_{\bar\rho}\subseteq\mathcal T_{\mathrm{box}}\), and these inclusions pass to subspaces.
1. On \(X\), \(\mathcal T_{\bar\rho}\subseteq\mathcal T_d\).
For \(\mathbf x,\mathbf y\in X\), \(\bar\rho(\mathbf x,\mathbf y)\le\sup_i\abs{x_i-y_i}\le d(\mathbf x,\mathbf y)\), so \(B_d(\mathbf x,\varepsilon)\subseteq B_{\bar\rho}(\mathbf x,\varepsilon)\).
2. On \(X\), \(\mathcal T_d\subseteq\mathcal T_{\mathrm{box}}\).
Given \(\mathbf x\in X\) and \(\varepsilon>0\), put \(\varepsilon_i=\varepsilon\,2^{-(i+1)/2}\), so \(\sum_i\varepsilon_i^2=\varepsilon^2/2\). For \(\mathbf y\) in the box neighborhood \(U=X\cap\prod_i(x_i-\varepsilon_i,x_i+\varepsilon_i)\), \(d(\mathbf x,\mathbf y)^2\le\sum_i\varepsilon_i^2<\varepsilon^2\), so \(U\subseteq B_d(\mathbf x,\varepsilon)\).
3. On \(\mathbb R^\infty\), \(\mathcal T_{\mathrm{prod}}\subsetneq\mathcal T_{\bar\rho}\subsetneq\mathcal T_d\subsetneq\mathcal T_{\mathrm{box}}\).
2. Each is strict because a sequence in \(\mathbb R^\infty\) converges to \(\mathbf 0\) in the coarser topology but not in the finer one; \(\mathbf e_n\) denotes the sequence with \(1\) in place \(n\) and \(0\) elsewhere.The sequence \(\mathbf e_n\) converges to \(\mathbf 0\) coordinatewise, but \(\bar\rho(\mathbf e_n,\mathbf 0)=1\).The sequence \(\mathbf y_n\) with its first \(n\) coordinates \(1/\sqrt n\) and the rest \(0\) has \(\bar\rho(\mathbf y_n,\mathbf 0)=1/\sqrt n\to0\), but \(d(\mathbf y_n,\mathbf 0)=1\).The sequence \(\mathbf w_n=\frac1n\mathbf e_n\) has \(d(\mathbf w_n,\mathbf 0)=\frac1n\to0\), but \(\mathbf w_n\) lies outside the box neighborhood \(\prod_i(-\frac1{i^2},\frac1{i^2})\) for every \(n\ge2\), since \(\frac1n\ge\frac1{n^2}\).
4. On \(H\), \(\mathcal T_{\mathrm{prod}}=\mathcal T_{\bar\rho}=\mathcal T_d\subsetneq\mathcal T_{\mathrm{box}}\).
Let \(\mathbf x\in H\) and \(\varepsilon>0\). Choose \(N\) with \(\sum_{i>N}1/i^2<\varepsilon^2/2\), and let \(W\) be the set of \(\mathbf y\in H\) with \(\abs{y_i-x_i}<\varepsilon/\sqrt{2N}\) for \(i\le N\), a basic product neighborhood of \(\mathbf x\) in \(H\). For \(\mathbf y\in W\), \(\abs{x_i-y_i}\le1/i\) for all \(i\) because \(x_i,y_i\in[0,1/i]\), so \[ d(\mathbf x,\mathbf y)^2<N\cdot\frac{\varepsilon^2}{2N}+\sum_{i>N}\frac1{i^2}<\varepsilon^2. \] Hence \(\mathcal T_d\subseteq\mathcal T_{\mathrm{prod}}\) on \(H\), and with step 1 the first three topologies agree. The set \(H\cap\prod_n[0,\frac1{n^2})\) is box-open in \(H\) and contains \(\mathbf 0\), but every basic product neighborhood of \(\mathbf 0\) in \(H\) leaves some coordinate \(n\ge2\) unrestricted and so contains the point \(\frac1n\mathbf e_n\notin\prod_n[0,\frac1{n^2})\).
5. Q.E.D.
Steps 1 and 2 prove (a), step 3 proves (b), and step 4 answers (c).