**I am in the process of typing a complete answer, but since it is getting late, I will only post some of the answer and I will update it tomorrow.**

Topological spaces where each point has a totally ordered local basis are known as $\textit{well-based}$ spaces. The notion of a well-based space is a generalization of the notion of a first countable space since the first countable spaces are the spaces where every point has a countable totally ordered local basis. A radial space is a topological space where $x\in\overline{A}$ iff there is some regular cardinal $\kappa$ and sequence $(a_{\alpha})_{\alpha<\kappa}$ of elements in $A$ where $(a_{\alpha})_{\alpha<\kappa}\rightarrow x$. Every well-based space is radial and every radial space is the quotient of a well-based space.

For example, if $X$ is a totally ordered set, then $X$ is well based in the order topology and in the lower limit topology.


$\textbf{When the sets $\mathscr{O}_{x}$ have the same cofinality}$

Suppose that $\kappa$ is a regular cardinal. Then we say that a space $X$ is $\kappa$-complete if whenever $|I|<\kappa$ and $O_{i}$ is open for each $i\in I$, then $\bigcap_{i\in I}O_{i}$ is also open. Completely regular $\kappa$-complete spaces are also called $P_{\kappa}$-spaces.

If $X$ is a topological space and $x\in X$, then define the character $\chi(X,x)$ of $X$ at $x$ to be the smallest cardinality of a subset $R\subseteq\mathscr{O}_{x}$ that generates $\mathscr{O}_{x}$.


Suppose that $X$ is a topological space. Then each $\mathscr{O}_{x}$ is generated by a chain of cofinality $\kappa$ if and only if $X$ is a $\kappa$-complete space where either $x$ is isolated or $\chi(X,x)=\kappa$ for each $x\in X$.
For example, the spaces where each $\mathscr{O}_{x}$ is generated by a chain of cofinality $\aleph_{0}$ are precisely the first countable spaces. It turns out that the $\kappa$-complete spaces $X$ where each $\mathscr{O}_{x}$ is generated by a chain of cofinality $\kappa$ behave very similar to the first countable spaces. In fact, the basic theory of $\kappa$-complete spaces is very similar to the basic theory of topological spaces since basic results about topological spaces often generalize the results about $\kappa$-complete spaces. The papers [1],[2] develop some of the basic theory of $\kappa$-complete spaces and generalize some facts about topological spaces to $\kappa$-complete spaces. 



1. [$\omega_{\mu}$-additive topological spaces][1]. Giuliano Artico; Roberto Moresco
Rendiconti del Seminario Matematico della Università di Padova (1982)
Volume: 67, page 131-141
ISSN: 0041-8994

2. [Remarks on some topological spaces of high power][1]. Roman Sikorski
Fundamenta Mathematicae (1950) Volume: 37, Issue: 1, page 125-136 ISSN: 0016-2736