I'm wondering in what context it makes sense to allow the dimension of a space to be an infinite ordinal instead of an infinite cardinal.
We would essentially have to be paying attention to relations on the dimension set in addition to its cardinality, such as a well-ordering -- for example, we have that $\mathbb{R}^{\omega+1}$ has 'dimension' $\omega$ if we only pay attention to cardinality since $|\omega+1|=\omega$, but it can naturally be considered to have dimension $\omega+1$ since there is no order preserving bijection between $\omega+1$ and $\omega$.
While the trivial answer to the above appears to be 'if the dimension set is well-ordered', the discussion here leads me to believe that there are situations where the dimension set does not inherently come with a well-ordering, but can be well-ordered to canonically produce an ordinal dimension value.
Further confounding things is the fact that in the presence of choice any class can be well-ordered, which would indicate that we could always use an ordinal instead of a cardinal if the above trivial answer is correct. In this case however we eventually run into difficulties with CH -- if $\mathbb{R}$ is the dimension set and we want to well-order it to determine the ordinal dimension, the ordinal we get depends on the answer to CH. Any assistance or references would be appreciated.
