# Topology of the space $C[0,\infty)$ of continuous functions.

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!

• They don't coincide – Bananach Apr 7 at 19:53
• @Bananach Thank you! – Viet Dang Apr 7 at 20:01

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 $$n$$.