Let $S$ and $P$ be the direct sum and direct product of countably many copies of $\mathbb R$. Clearly, $P$ should have the product topology. On $S$, we could consider two different topologies:
the subspace topology given by the canonical inclusion $S \to P$,
the cellular topology, given by considering $\mathbb R$ as a CW complex (say, with the $0$-cells being $\mathbb Z$), and identifying $S$ with the direct limit of $0 \to \mathbb R \to \mathbb R^2 \to \mathbb R^3 \to \dots$.
Do these topologies agree?
No, the topology in (2) is finer. For instance, the set $\{(f_n)_{n\in\Bbb N}\,:\,\forall n, \lvert f_n\rvert<2^{-n}\}$ isn't open in (1), but it is in (2): notice that its intersection with each $\Bbb R^n\times 0^{\{m> n\}}$ is open. As it turns out, topology (2) is the subspace topology induced by the box topology.