Every first countable space is clearly well-based. Furthermore, if $X$ is a totally ordered set, then $X$ is well based in the order topology and in the lower limit topology, and as long as $\{x\in X:x<x_0\},\{x\in X:x>x_0\}^*$ (the $Z^*$ means $Z$ with the reversed ordering) always have the same cofinality or one of these sets has a maximum element or is empty, then $X$ must be well-based in the order topology as well.
Every first countable space is clearly well-based. Furthermore, if $X$ is a totally ordered set, then $X$ is well based in the order topology and in the lower limit topology.
Every first countable space is clearly well-based. Furthermore, if $X$ is a totally ordered set, then $X$ is well based in the in the lower limit topology, and as long as $\{x\in X:x<x_0\},\{x\in X:x>x_0\}^*$ (the $Z^*$ means $Z$ with the reversed ordering) always have the same cofinality or one of these sets has a maximum element or is empty, then $X$ must be well-based in the order topology as well.
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. The papers 3,4 mention well-based spaces, but I have not found any other mathematical literature that mentions well-based spaces and calls these spaces well-based.
For exampleEvery first countable space is clearly well-based. Furthermore, 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}$$\textbf{When the sets $\mathscr{O}_{x}$ have the same cofinality $\kappa$}$
The well-based spaces where each $\mathscr{O}_{x}$ has the same cofinality seem to have a special importance among all the well-based spaces.
Suppose that $\kappa$ is a regular cardinal. Then define a $\kappa$-well based space to be a well based space where each $\mathscr{O}_{x}$ has a cofinal chain of cofinality $\kappa$. Even though the notion of a $\kappa$-well based space seems like a natural notion to study, I have not found any papers that mention $\kappa$-well based spaces. The notion of a $\kappa$-well based space is the correct generalization of the notion of a separable space to $\kappa$-complete spaces.
Suppose that $\kappa$ is a regular cardinal. Then we say that a space $X$ is $\kappa$-complete if whenever $|I|<\kappa$ and $O_{i}$ is open for each $i\in I$, then $\bigcap_{i\in I}O_{i}$ is also open. Completely regular $\kappa$-complete spaces are also called $P_{\kappa}$-spaces.
Suppose that $X$ isClearly a topological space. Then each $\mathscr{O}_{x}$$X$ is generated by a chain of cofinality $\kappa$ if and only if-well based iff $X$ is a $\kappa$-complete space where either $x$ is isolated or $\chi(X,x)=\kappa$ for each $x\in X$. For For example, the first countable spaces are precisely the $\kappa$-well based spaces.
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, ad the notion of a $\kappa$-well based space is the correct way to generalize the notion of a first countable space to $\kappa$-complete spaces. The papers [1]1,[2]2 develop some of the basic theory of $\kappa$-complete spaces and generalize some facts about topological spaces to $\kappa$-complete spaces.
$\kappa$-complete spaces can easily be generated by topological spaces. Suppose that $X$ is a topological space and $\kappa$ is a regular cardinal. Then define $(X)_{\kappa}$ to be the topological space with the same underlying set as $X$ but where $(X)_{\kappa}$ is generated by the basis consisting of sets of the form $\bigcap_{i\in I}U_{i}$ where each $U_{i}$ is open in $X$ and where $|I|<\kappa$. Then $(X)_{\kappa}$ is always a $\kappa$-complete space.
The following results are a rather straightforward generalizations of results about first countable spaces, but they illustrate that the notion of a $\kappa$-well based space is the correct generalization of the notion of a first countable space.
$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-well based space. If $A\subseteq X$, then $x\in\overline{A}$ if and only if there is some sequence $(a_{\alpha})_{\alpha<\kappa}$ with $a_{\alpha}\in A$ for each $\alpha<\kappa$ converging to $x$.
$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-well based space. Let $Y$ be a topological space and let $f:X\rightarrow Y$ be a function. Then the following are equivalent.
$f:X\rightarrow Y$ is a continuous function.
$f:X\rightarrow(Y)_{\kappa}$ is a continuous function.
Whenever $(x_{\alpha})_{\alpha<\kappa}\rightarrow x$, we have $(f(x_{\alpha}))_{\alpha<\kappa}\rightarrow f(x).$
$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-complete space. Then the following are equivalent.
Every open cover $\mathcal{U}$ of cardinality $\kappa$ has a subcover of cardinality less than $\kappa$.
Every sequence $(x_{\alpha})_{\alpha<\kappa}$ accumulates at some point.
$\mathbf{Theorem}$ Let $X$ be a $\kappa$-well based space. If $(x_{\alpha})_{\alpha<\kappa}$ is a sequence that accumulates at some point $x$, then there exists a subsequence $(x_{j(\alpha)})_{\alpha<\kappa}$ where $j:\kappa\rightarrow\kappa$ is strictly increasing but where $x_{j(\alpha)}\rightarrow x$.
$\mathbf{Theorem}$ Suppose $X$ is a $\kappa$-well based space. Then the following are equivalent.
If $\mathcal{U}$ is an open cover with cardinality $\kappa$, then $\mathcal{U}$ has a subcover of cardinality less than $\kappa$.
If $(x_{\alpha})_{\alpha<\kappa}$ is a sequence, then there is some strictly increasing map $j:\kappa\rightarrow\kappa$ such that $(x_{j(\alpha)})_{\alpha<\kappa}$ converges to some point.
We therefore conclude that the $\kappa$-well based spaces are spaces that one can study simply by considering the $\kappa$-sequences and their convergence. Furthermore, one can get by with thinking about $\kappa$-well based spaces n the same way that one thinks about first countable spaces.
I hope I did not stray too far from the information which you are looking for in this answer.
$\mathbf{References}$
[$\omega_{\mu}$-additive topological spaces][1]$\omega_{\mu}$-additive topological spaces. Giuliano Artico; Roberto Moresco Rendiconti del Seminario Matematico della Università di Padova (1982) Volume: 67, page 131-141 ISSN: 0041-8994
[Remarks on some topological spaces of high power][1]Remarks on some topological spaces of high power. Roman Sikorski Fundamenta Mathematicae (1950) Volume: 37, Issue: 1, page 125-136 ISSN: 0016-2736
Convergence properties and compactifications. Robert Leek. University of Oxford
An internal characterisation of radiality. Robert Leek. University of Oxford
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.
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.
[$\omega_{\mu}$-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
[Remarks on some topological spaces of high power][1]. Roman Sikorski Fundamenta Mathematicae (1950) Volume: 37, Issue: 1, page 125-136 ISSN: 0016-2736
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. The papers 3,4 mention well-based spaces, but I have not found any other mathematical literature that mentions well-based spaces and calls these spaces well-based.
Every first countable space is clearly well-based. Furthermore, 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 $\kappa$}$
The well-based spaces where each $\mathscr{O}_{x}$ has the same cofinality seem to have a special importance among all the well-based spaces.
Suppose that $\kappa$ is a regular cardinal. Then define a $\kappa$-well based space to be a well based space where each $\mathscr{O}_{x}$ has a cofinal chain of cofinality $\kappa$. Even though the notion of a $\kappa$-well based space seems like a natural notion to study, I have not found any papers that mention $\kappa$-well based spaces. The notion of a $\kappa$-well based space is the correct generalization of the notion of a separable space to $\kappa$-complete spaces.
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.
Clearly a topological space $X$ is $\kappa$-well based iff $X$ is a $\kappa$-complete space where either $x$ is isolated or $\chi(X,x)=\kappa$ for each $x\in X$. For example, the first countable spaces are precisely the $\kappa$-well based spaces.
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, ad the notion of a $\kappa$-well based space is the correct way to generalize the notion of a first countable space to $\kappa$-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.
$\kappa$-complete spaces can easily be generated by topological spaces. Suppose that $X$ is a topological space and $\kappa$ is a regular cardinal. Then define $(X)_{\kappa}$ to be the topological space with the same underlying set as $X$ but where $(X)_{\kappa}$ is generated by the basis consisting of sets of the form $\bigcap_{i\in I}U_{i}$ where each $U_{i}$ is open in $X$ and where $|I|<\kappa$. Then $(X)_{\kappa}$ is always a $\kappa$-complete space.
The following results are a rather straightforward generalizations of results about first countable spaces, but they illustrate that the notion of a $\kappa$-well based space is the correct generalization of the notion of a first countable space.
$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-well based space. If $A\subseteq X$, then $x\in\overline{A}$ if and only if there is some sequence $(a_{\alpha})_{\alpha<\kappa}$ with $a_{\alpha}\in A$ for each $\alpha<\kappa$ converging to $x$.
$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-well based space. Let $Y$ be a topological space and let $f:X\rightarrow Y$ be a function. Then the following are equivalent.
$f:X\rightarrow Y$ is a continuous function.
$f:X\rightarrow(Y)_{\kappa}$ is a continuous function.
Whenever $(x_{\alpha})_{\alpha<\kappa}\rightarrow x$, we have $(f(x_{\alpha}))_{\alpha<\kappa}\rightarrow f(x).$
$\mathbf{Theorem}$ Suppose that $X$ is a $\kappa$-complete space. Then the following are equivalent.
Every open cover $\mathcal{U}$ of cardinality $\kappa$ has a subcover of cardinality less than $\kappa$.
Every sequence $(x_{\alpha})_{\alpha<\kappa}$ accumulates at some point.
$\mathbf{Theorem}$ Let $X$ be a $\kappa$-well based space. If $(x_{\alpha})_{\alpha<\kappa}$ is a sequence that accumulates at some point $x$, then there exists a subsequence $(x_{j(\alpha)})_{\alpha<\kappa}$ where $j:\kappa\rightarrow\kappa$ is strictly increasing but where $x_{j(\alpha)}\rightarrow x$.
$\mathbf{Theorem}$ Suppose $X$ is a $\kappa$-well based space. Then the following are equivalent.
If $\mathcal{U}$ is an open cover with cardinality $\kappa$, then $\mathcal{U}$ has a subcover of cardinality less than $\kappa$.
If $(x_{\alpha})_{\alpha<\kappa}$ is a sequence, then there is some strictly increasing map $j:\kappa\rightarrow\kappa$ such that $(x_{j(\alpha)})_{\alpha<\kappa}$ converges to some point.
We therefore conclude that the $\kappa$-well based spaces are spaces that one can study simply by considering the $\kappa$-sequences and their convergence. Furthermore, one can get by with thinking about $\kappa$-well based spaces n the same way that one thinks about first countable spaces.
I hope I did not stray too far from the information which you are looking for in this answer.
$\mathbf{References}$
$\omega_{\mu}$-additive topological spaces. Giuliano Artico; Roberto Moresco Rendiconti del Seminario Matematico della Università di Padova (1982) Volume: 67, page 131-141 ISSN: 0041-8994
Remarks on some topological spaces of high power. Roman Sikorski Fundamenta Mathematicae (1950) Volume: 37, Issue: 1, page 125-136 ISSN: 0016-2736
Convergence properties and compactifications. Robert Leek. University of Oxford
An internal characterisation of radiality. Robert Leek. University of Oxford
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.
[$\omega_{\mu}$-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
[Remarks on some topological spaces of high power][1]. Roman Sikorski Fundamenta Mathematicae (1950) Volume: 37, Issue: 1, page 125-136 ISSN: 0016-2736