This question came to mind upon M W's answer to M W's answer to thisthis recent Math.SE question - the question asked about finite spaces, but it was pointed out that all that was really needed was the Alexandrov property, the class of spaces for which the intersection of an arbitrary collection of open sets is open.
added link to M W's answer for clarity, just in case the user's displayname changes, etc.
The Amplitwist
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