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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

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

Question. Is this Lindelöf topology?

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    $\begingroup$ 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) $\endgroup$ Oct 29, 2020 at 16:28
  • $\begingroup$ Thank you. What about $\sigma$ compact topology? $\endgroup$
    – solver6
    Oct 29, 2020 at 16:40
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    $\begingroup$ 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. $\endgroup$ Oct 29, 2020 at 18:14
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    $\begingroup$ This seems vaguely like dealing with the box topology instead of the product topology, and probably runs into similar issues. $\endgroup$
    – LSpice
    Oct 29, 2020 at 19:16
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    $\begingroup$ 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 $\mathbf{C}((t))$. $\endgroup$ Nov 1, 2020 at 8:53

1 Answer 1

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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.

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