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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 15th 2017

the entry neighbourhood base was in a funny state. I have edited a little. No mention of filters yet.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeMay 15th 2017

I edited a little myself.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 15th 2017

That makes sense, thanks.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeMay 15th 2017
• (edited May 15th 2017)

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.).

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 15th 2017

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).

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeMay 31st 2017

Not requiring the neighbourhoods to be open, in my mind, helps to remember that they're supposed to be neighbourhoods of something. (Because otherwise there's no restriction at all!) There's open sets, and then there's neighbourhoods.