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the entry neighbourhood base was in a funny state. I have edited a little. No mention of filters yet.
I edited a little myself.
That makes sense, thanks.
I edited some more: I removed the superscripts $x$ as superfluous (and overly busy as notation), and removed the word “open” preceding the $U_i$ since there is no insistence on that (as when one says, “a base of compact neighborhoods”, etc.).
Sure. I had added those superscripts to amplify that the base elements are all required to be neighbourhoods of $x$, too, instead of more general open subsets. I sometimes see that condition being omitted (forgotten).
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