Firstly, I want to emphasise that I understand why the open set version of the axioms of topology require that only only the finite intersection of open sets is guaranteed to be open. I know some standard demonstrations of this necessity, like that on the real line $\forall n \in \mathbb{N}^+, (-1/n, 1/n)$ is open but $\bigcap_{n \in \mathbb{N}^+} (-1/n, 1/n) = \{0\}$ which is not open. This is not what my question is about.
However, I have never been pleased with the asymmetry between the axioms for intersection and union, I have always found it kinda ugly and inelegant. The worst presentations explicitly invoke indexing by natural numbers, but the neatest versions I've seen are still asymmetric, saying that the intersection of two open sets is open, but union of arbitrary open sets is open.
My question is whether there is an alternate axiomatisation of topology via open sets (potentially radically different) which is aesthetically "nicer" in this regard, which removes the explicit asymmetry while still being logically equivalent.
I am familiar with other axiomatisations of topology which do not use open sets at all, such as the neighborhood definition or the Kuratowski closure axioms. While these are interesting, they are not what this question is about (although do share if you think you have a really strange and interesting formalism).
An Alexdrandov space is a topological space for which the
intersection of any collection of open sets is open.
A $T_1$ Alexdrandov space is discrete.
That is what happens when the axioms are made symmetrical.
A bunch of low level spaces not fit for analysis,