In Fourier Analysis by Javier Duoandikoetxea, a function $f:\mathbb{R}^n\to\mathbb{R}$ is said to be in the Schwarz Class $\mathcal{S}(\mathbb{R}^n)$ if it is infinitely differentiable and if all of its derivatives decrease rapidly at infinity; i.e., if for all $\alpha,\beta\in\mathbb{N}^n$ we have $$p_{\alpha, \beta}(f) := \sup_{x\in\mathbb{R}^n}\lvert x^\alpha D^\beta f(x)\rvert < \infty,$$ where $x^\alpha = x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ and $$D^\beta f = \frac{\partial^{\beta_1 + \cdots + \beta_n}}{\partial x_1^{\beta_1}\cdots\partial x_n^{\beta_n}}.$$
The author then defined a topology on $\mathcal{S}(\mathbb{R}^n)$ as such: "The collection $\{p_{\alpha,\beta}\}$ is a countable family of seminorms on $\mathcal{S}(\mathbb{R}^n)$, and we can use it to define a topology on $\mathcal{S}(\mathbb{R}^n)$: a sequence $\{\phi_k\}$ converges to $0$ if and only if for all $\alpha,\beta\in\mathbb{N}^n$, $$\lim_{k\to\infty}p_{\alpha,\beta}(\phi_k) = 0." $$
I don't understand how this gives a topology on the Schwarz Class. In general, a set $X$ can be endowed with two (non-homeomorphic) topologies $T, T'$ such that a sequence $\{x_n\}$ converges in $(X, T)$ if and only if it converges in $(X, T')$, so how does just specifying convergent sequences define a topology? What are the open sets?