A classic text is Dimension Theory by Hurevicz & Wallman (1941)

Among the types of dimension for topological spaces are the "upper inductive dimension" and the "lower inductive dimension". The form of the definition is such that they could be adapted to something with ordinal values. As I recall (it has been many years) Hurewicz & Wallman discuss the literature on this possibility in the notes at the end of one of their chapters.

Like this... For a topological space $X$ we say $\text{ind}(X) \le k$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) < k$. With $\text{ind}( \varnothing) = -1$ this is conventionally used to define the small inductive dimension for natural numbers $k$. $\text{ind}(X) = k$ iff $k$ is the least natural number such that $\text{ind}(X) \le k$.

But it will also work for transfinite ordinals. Because of the inductive form of the definition.

$\text{ind}(X) \le \omega$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) < \omega$. $\text{ind}(X) \le \omega+1$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) \le \omega$.

Similar discussion for the large inductive dimension.

How interesting this turns out to be, I do not recall.

A classic text is Dimension Theory by Hurevicz & Wallman (1941)

Among the types of dimension for topological spaces are the "upper inductive dimension" and the "lower inductive dimension". The form of the definition is such that they could be adapted to something with ordinal values. As I recall (it has been many years) Hurewicz & Wallman discuss the literature on this possibility in the notes at the end of one of their chapters.

Like this... For a topological space $X$ we say $\text{ind}(X) \le k$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) \le k-1$. With $\text{ind}( \varnothing) = -1$ this is conventionally used to define the small inductive dimension for natural numbers $k$. $\text{ind}(X) = k$ iff $k$ is the least natural number such that $\text{ind}(X) \le k$.

But it will also work for transfinite ordinals. Because of the inductive form of the definition.

$\text{ind}(X) \le \omega$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) < \omega$. $\text{ind}(X) \le \omega+1$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) \le \omega$. And in general, $\text{ind}(X) \le \gamma$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) < \gamma$.

Similar discussion for the large inductive dimension.

How interesting this turns out to be, I do not recall.

A classic text is Dimension Theory by Hurevicz & Wallman (1941)

Among the types of dimension for topological spaces are the "upper inductive dimension" and the "lower inductive dimension". The form of the definition is such that they could be adapted to something with ordinal values. As I recall (it has been many years) Hurewicz & Wallman discuss the literature on this possibility in the notes at the end of one of their chapters.

Like this... For a topological space $X$ we say $\text{ind}(X) \le k$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal U$, $\text{ind}(\partial V) < k$. With $\text{ind}( \varnothing) = -1$ this is conventionally used to define the small inductive dimension for natural numbers $k$. But it will also work for transfinite ordinals. Because of the inductive form of the definition.

Similar discussion for the large inductive dimension.

How interesting this turns out to be, I do not recall.

A classic text is Dimension Theory by Hurevicz & Wallman (1941)

Among the types of dimension for topological spaces are the "upper inductive dimension" and the "lower inductive dimension". The form of the definition is such that they could be adapted to something with ordinal values. As I recall (it has been many years) Hurewicz & Wallman discuss the literature on this possibility in the notes at the end of one of their chapters.

Like this... For a topological space $X$ we say $\text{ind}(X) \le k$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) < k$. With $\text{ind}( \varnothing) = -1$ this is conventionally used to define the small inductive dimension for natural numbers $k$. $\text{ind}(X) = k$ iff $k$ is the least natural number such that $\text{ind}(X) \le k$.

But it will also work for transfinite ordinals. Because of the inductive form of the definition.

$\text{ind}(X) \le \omega$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) < \omega$. $\text{ind}(X) \le \omega+1$ iff there is a base $\mathcal B$ for the topology such that for all $V \in \mathcal B$, $\text{ind}(\partial V) \le \omega$.

Similar discussion for the large inductive dimension.

How interesting this turns out to be, I do not recall.