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Consider the space $C[0,\infty)$ of continuous functions on $[0,\infty)$. Consider the following two topologies.

The first topology, denoted $\tau_1$ is generated by cylinder sets, i.e. sets of the form: $\{\omega\in C[0,\infty) \text{ such that }(\omega_{t_1},..\omega_{t_2})\in A_1\times...\times A_n\}$, where $A_1,..A_n$ are open sets in $\mathbb{R}$.

The second topology, denoted $\tau_2$ is the metric topology induced by the following metric: $d(\omega_1,\omega_2) = \sum_{n=1}^\infty\frac{1}{2^n}\sup_{0\leq t \leq n}(|\omega_1(t)-\omega_2(t)|\wedge 1)$.

What I'm trying to answer is whether the two topologies coincide. I have managed to show that $\tau_1\subseteq\tau_2$. I'm not sure how to proceed with the other direction.

Any hint or direction or reference material would be greatly appreciated.

Thanks!

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    $\begingroup$ They don't coincide $\endgroup$ – Bananach Apr 7 at 19:53
  • $\begingroup$ @Bananach Thank you! $\endgroup$ – Viet Dang Apr 7 at 20:01
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They do not coincide. For instance, pointwise convergence implies convergence in $\tau_1$, but not in $\tau_2$.

For example, let $h(x)$ be $1$ on $[2; 3]$ and $0$ on $[0; 1] \cup [4; \infty)$. Let $f_n(x) = h(\frac{x}{n})$. Then $f_n \to_{\tau_1} 0$ (as for any $x$ for all sufficiently large $n$ we have $f_n(x) = 0$), but $d(f_n, 0) \geqslant \frac{1}{2}$ for all $n$.

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  • $\begingroup$ That was very helpful, thank you! $\endgroup$ – Viet Dang Apr 7 at 20:03
  • $\begingroup$ @VietDang So, why not accepting this answer? $\endgroup$ – Jochen Apr 9 at 7:49
  • $\begingroup$ @Jochen I did now. I'm not very familiar with the site. $\endgroup$ – Viet Dang Apr 16 at 19:52

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