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I added a synthetic definition of open subspace due to Penon.
I just woke up, but should that sentence end with $x \in U$ instead of $y \in U$?
Yes, it should, thanks.
That is highly reminiscent of the topology defined by an apartness relation. This topology is always $T_1$, although I don't see why it would need to be $T_2$.
The first sentence of open subspace defined a subset to be open if its inclusion map was an open map. This is a bit circular given that the first sentence of open map defines a map to be open if the image of each open subspace is open. (-: I rearranged the page to start with the classical notion as part of the definition of a topological space and then proceed to generalizations to convergence spaces, locales, and so on.
That first sentence wasn't supposed to be a definition (despite being labelled as such by Urs) but an all-encompassing true statement going to the heart of the matter. If anything should be rearranged, it's open map. That said, your new version of the first sentence of open subspace (also not a definition) is actually pretty good.
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