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