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Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of $\mathbb{R}$ with respect to $S$, i.e.

$\mathbb{R}(S,\mu):=\mathbb{R}^S/\sim$, where two maps $f,g$ are called equivalent, if $\mu(\{i\in S| f(i)=g(i)\})=1$. This is again an ordered field. Equip it with the topology generated by

$\{B_\varepsilon(x)|x\in \mathbb{R}(S,\mu),\varepsilon \in \mathbb{R}(S,\mu),\varepsilon>0\}$.

Then my question is: What is the smallest cardinality of a local basis around $0$ depending on the cardinality of |S| and (possibly) on the ultrafilter?

Examples:

If $S=\{pt\}$, we get $\mathbb{R}(S,\mu)\cong \mathbb{R}$ and hence it has a countable base for the topology. In the case $S=\mathbb{N}$ and a non-principal ultrafilter one can show, that there is no countable base for the topology (saturation argument) and it is at most $|\mathbb{R}(\mathbb{N},\mu)|=|\mathbb{R}|$. CH would tell us, that it is $|\mathbb{R}|$. But maybe there is a good (avoiding CH) reason, why it has exactly this cardinality.

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up vote 2 down vote accepted

Your question is essentially equivalent to the one here; for what is known see in particular the bottom of Joel David Hamkins' answer there.

To see the equivalence, note that choosing such a local base $B$ at zero is equivalent (at least assuming AC) to choosing a set $E$ of positive elements of $\mathbb{R}(S,\mu)$ such that for all $x>0$ there is a $y\in E$ with $y < x$. Given such a set $E$ we can take $B = \{(-x,x) \mid x\in E\}$. Given a basis $B$ we can take $E$ to contain one element $x$ for each neighborhood in $B$, with $x$ chosen so $(-x,x)$ is contained in the neighborhood.

Choosing $E$ is in turn equivalent to choosing a cofinal set $F$, i.e. one which contains a $y>x$ for any $x\in\mathbb{R}(S,\mu)$. You can pass between $E$ and $F$ by taking the reciprocal of all the elements.

The cofinality of $\mathbb{R}(S,\mu)$ is the smallest such cofinal set, so this is what you are asking for. According to Joel's answer, these are studied in Shelah's Possible Cofinality (PCF) theory.

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