Intuitively, I understand the construction of the hyperreals by ultraproducts as a process of turning the limit operation into an algebraic object. More precisely, to check the existence of the limit $\lim_{x\to a} f(x)$, is equivalent to calculate $\overline f(\tilde a)$ for every $\tilde a$ such that $\tilde a\neq a$ and, $a-u<\tilde a<a+u,\ \forall u>0$, and check if its projection on $\mathbb R$ is the same for every $\tilde a$ of that form; and if it does, then the common value is the limit.

I'd like to work with this kind of tecnique, but for general topological spaces. However, it is not that simple when the topological space has points of different character. Like, a point that admits countable local basis and other point that does not admit it.

When we construct the hyperreals we take a non-principal ultrafilter $\mathcal U\subset\mathcal P(\mathbb N)$ containing the cofinite subsets of $\mathbb N$. The convergence with respect to this ultrafilter is equivalent to the convergence of a sequence in the classical sense. Since every point in $\mathbb R$ has a local basis $\mathcal B$ such that $(\mathcal B,\supset) \equiv(\mathbb N,\leq)$, the ultrapower of $\mathbb R$ by $\mathcal U\subset\mathcal P(\mathbb N)$ "translates" the convergence to every point $a\in\mathbb R$ via the elements $\tilde a$ such that $\tilde a\neq a$ and, $a-u<\tilde a<a+u,\ \forall u>0$, as I said in the first paragraph. This kind of idea should work analogously in any first countable topological space.

But what about topological spaces that are not first countable? I know, a priori, that if every point $x\in X$ has a local basis $\mathcal B$ such that $(\mathcal B,\supset) \equiv(I,\leq)$, for a common directed set $(I,\leq)$, then, we may pick an ultrafilter $\mathcal U\subset \mathcal P(I)$ containing the subsets of the form $S_i=\{j\in I:i\leq j\}$ and construct the ultrapower of $X$ by $\mathcal U$. This construct works in the way I'd like to. However, it relies on the hypothesis that

every point $x\in X$ has a local basis $\mathcal B$ such that $(\mathcal B,\supset) \equiv(I,\leq)$, for a common directed set $(I,\leq)$.

I'd like to know if there is a construction like that for any topological space $X$, or at least for a good class of them, say compact or locally compact and Hausdorff, not requiring this hypothesis.