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skulblaka@sh.itjust.works ⁨1⁩ ⁨month⁩ ago

Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection, that the open sets be extensional, and that the membership predicate (of points in open sets) be two-valued. The definition of continuous function remains unchanged other than having to be worded carefully to continue to make sense after these generalizations.

I know what most of these words mean, individually. But I do not know what this article is attempting to communicate to me.

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Septimaeus@infosec.pub ⁨1⁩ ⁨month⁩ ago
A chu space is a “binary relation” (aka mapping/function) between two sets. Imagine an excel table.

Dropping “closure” requirement means a point in the set needn’t be reachable from every other point. Imagine a game of snake, where walling yourself in is possible.

“Extensional” means each point extends (originates from) another. Imagine leaving a trail of breadcrumbs so you can retrace your steps.

“Two-valued:” refers to the two sets (see 1), since both are needed to avoid ambiguous extension (see 3). Now instead of breadcrumbs, imagine using a string to retrace your steps, so if you cross an old path you know which one to take.

Defined dynamically, a chu space is a complementary pair of “continuous functions.” Imagine again the game, where the snake’s position at any frame/tick can be equivalently reproduced by either “everywhere its head has been + its next direction” OR “ the current position of its head + every direction it has moved.”
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