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


$V\_{n,v,\varepsilon} = \{f=a\_kt^k+\ldots+a\_0+a\_1t+\ldots\in \mathbb{C}(t)| \sum\_{k\leq m<n}|m|^v|a\_m|<\varepsilon\}$



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> Question. Is this Lindelöf topology?
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What is good $t$-adic like topology on $\mathbb{C}(t)$?
Let $\|f\|\_{n, m} = \sum\_{i<n}(|i|^m+10)|a\_i|$ 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.


