# What is good $t$-adic like topology on $\mathbb{C}(t)$?

Each function $$f\in\mathbb{C}(t)$$ can be rewritten in the form $$f = a_{k}t^{k}+\ldots+a_0+a_1t+\ldots$$, $$k\in\mathbb{Z}$$ and it is possible to define the topology with the open prebase at zero

$$V_{n,v,\varepsilon} = \{f=a_kt^k+\ldots+a_0+a_1t+\ldots\in \mathbb{C}(t)| \sum_{k\leq m

Question. Is this Lindelöf topology?

• I don't think there's a natural locally compact topology on $\mathbb{C}(t)$: the fields admitting such a thing are called local fields and are completely classified (this might be possible to fix by asking for a weaker compatibility of the topology with the field structure, but I'm personally .skeptical) Oct 29, 2020 at 16:28
• Thank you. What about $\sigma$ compact topology? Oct 29, 2020 at 16:40
• Each one of your prebasic open sets is $t$-adically open (the set contains the open disk of radius $p^{-n}$ around each of its points). But for every $n$, the $t$-adic disk of radius $p^{-n}$ centered at $0$ contains $V_{n+1,n+2,\{1\}}$. So it seems to me that this topology is the same as the $t$-adic topology. Oct 29, 2020 at 18:14
• This seems vaguely like dealing with the box topology instead of the product topology, and probably runs into similar issues. Oct 29, 2020 at 19:16
• What if we consider the inverse limit of topological rings $\mathbf{C}[t]/t^n$ each with the Euclidean topology? This gives a mixture of the $t$-adic and Euclidean topologies on $\mathbf{C}[[t]]$. I didn't check, but it may be that this topology does not extend to a ring topology $\mathbf{C}((t))$. Nov 1, 2020 at 8:53

Let $$\|f\|_{n, m} = \sum_{i denote a family of seminorms on $$\mathbb{C}(t)$$. Note that for any $$\{(n_v, m_v)\}_{v\text{ in finite subset of }\mathbb{N}}$$ there is a seminorm $$\|.\|_{n,m}$$ such that $$\|f\|_{n, m}\geq\sup_v\|f\|_{n_v, m_v}$$. Also these seminorms define the same topology as stated in the problem. Resulting topological space is separabel so for some countable set of $$F_i\in\mathbb{C}(t)$$ this set is dense. Let $$\{U_{\lambda}\}$$ be some open cover of this topological space. For each $$F_i$$ and $$(n, m)$$ define $$\varepsilon_{i, n, m}$$ as supremum of $$\varepsilon >0$$ such that the $$\varepsilon$$ ball around $$F_i$$ in $$\|.\|_{n, m}$$ lies inside of $$U_{\lambda}$$ for some $$\lambda$$ and let $$\{\|f\|_{n, m}<\varepsilon_{i, n, m}/2\}\subset U_{\lambda_{i, n, m}}$$ in result $$\cup_{i, n, m}U_{\lambda_{i, n, m}}$$ is countable open subcover.