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Here's Prob. 8, Sec. 20 in the book Topology by James R. Munkres, 2nd edition:

Let $X$ be the subset of $\mathbb{R}^\omega$ consisting of all sequences $x$ such that $\sum x_i^2$ converges. Then the formula $$d(x,y) = \left[ \sum_{i=1}^\infty \left( x_i - y_i \right)^2 \right]^{\frac{1}{2}} $$ defines a metric on $X$. 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.

(a) Show that on $X$, we have the inclusions $$ \mbox{ box topology } \ \supset \ \ell^2 \mbox{ topology } \ \supset \ \mbox{ uniform topology}.$$

(b) 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 $$H = \prod_{n \in \mathbb{Z}_+} \left[0, \frac{1}{n} \right]$$ is contained in $X$; it is called the Hilbert cube. Compare the four topologies that $H$ inherits as a subspace of $X$.

My effort:

Part (a):

Let $x \in X$ and $\varepsilon > 0$. Let $y$ be a point of the open ball of radius $\varepsilon$ in the $d$-metric space centered at $x$. Then we can find a real number $\delta > 0$ such that the open ball of radius $\delta$, in the $d$-metric space again, and centered at $y$ is contained in the former open ball; that is, $$B_d (y, \delta) \subset B_d ( x, \varepsilon).$$ Now let $$B = X \cap \left[ \prod_{n \in \mathbb{N}} \left( y_n - \frac{\delta}{2^n}, y_n + \frac{\delta}{2^n} \right) \right].$$ Then $y \in B$ and if $z \in B$, then we have $\left\vert y_n - z_n \right\vert < \frac{\delta}{2^n}$ for all $n$ and so $$d(y,z) \leq \left[ \sum_{n=1}^\infty \frac{\delta^2}{4^n} \right]^{\frac{1}{2}} = \frac{\delta}{\sqrt{3}} < \delta,$$ showing that $z \in B_d(y, \delta)$ and hence that $z \in B_d(x, \varepsilon)$.

Thus, for each point $x \in X$ and each real number $\varepsilon > 0$, and for each point $y \in B_d(x, \varepsilon)$, we can find a basis element $B$ for the subspace topology inherited by $X$ from the box topology on $\mathbb{R}^\omega$ such that $y \in B \subset B_d(x, \varepsilon)$, which shows the first inclusion. Am I right?

Now let's suppose $\varepsilon < 1$. Let $\delta$ be any real number such that $0 < \delta < \varepsilon$. Then, if $y \in B_d(x, \delta)$, we have $$\left\vert x_n - y_n \right\vert \leq d(x,y) < \delta$$ for all $n$, from which it follows that $\tilde{\rho}(x,y) \leq \delta < \varepsilon$.

Thus, for each point $x \in X$ and for each real number $\varepsilon > 0$, we can find a real number $\delta > 0$ such that $$ B_d(x, \delta) \subset B_{\tilde{\rho}}(x, \varepsilon),$$ which is equivalent to the second inclusion. Am I right?

How to solve the remaining parts of the problem?

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  • $\begingroup$ The usual meaning of the Hilbert cube is $H$ with the (Tychonoff) product topology.... If $T$ and $T'$ are topologies on a set $S$, one way to show that $T\subset T'$ is to take some (any) bases $B, B'$ for $T,T'$ respectively, and show that whenever $p\in b\in B,$ there exists $b'\in B'$ such that $p\in b'\subset b.$ $\endgroup$ Feb 17 '20 at 0:06
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For the last inclusion of part (a),use the fact that $\overline{\rho}(\mathbf x,\mathbf y)\leq d(\mathbf x,\mathbf y)$.

For part (b), Look at the set $\mathbf U=\underset{n\in \mathbb N}\Pi (\frac{-1}{n+1},\frac{1}{n+1})$,whose intersection with $\mathbb R^{\infty}$ is open in box topology of $\mathbb R^{\infty}$,contains $\mathbf 0$,but does not contain $\mathbf B_{l^2}(0,\epsilon)$ $\forall \epsilon>0 $,by looking at the element $(0,0,0,......,0,\epsilon/2,0,0,0,.......)$,where $\epsilon/2$ sits in sufficiently large co-ordinate.This shows that box topology is strictly finer than the $l^2$ topology.

Similiarly, $(\delta/2,\delta/2,......,\delta/2,0,0,0,0,.......)\in (\mathbf B_{\overline{\rho}}(0,\delta)-\mathbf B_{l^2}(0,\epsilon))\cap \mathbb R^{\infty}$ $\forall \epsilon>0$ and $\delta>0$ if $\delta/2$ is repeated sufficiently many times.

Now,you can carry on similar arguments.

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