Showing that two topologies are the same

Question

Let $G$ be an abelian group and suppose that it is a topological group (at least T1) with respect two topologies $\tau_1$ and $\tau_2$.

Assume that $G$ admits the following sequence of subgroups $\{C_r\}_{r\in\mathbb Z}$:

$$\bigcup_rC_r =G\supset\ldots\supset C_r\supset C_{r+1}\supset \ldots C_0\supset C_1\supset C_2\supset\ldots \supset 0=\bigcap_rC_r\,.$$

If each $C_r$ is closed in both topologies $\tau_1$ and $\tau_2$, can we deduce that $\tau_1=\tau_2$?

Answer 1

No. Let $G=\Bbb R^{\Bbb Z}$, $C_r=\Bbb R^{\Bbb Z\cap[r,\infty)}$, $\tau_1$ the product topology, $\tau_2$ the discrete topology.