Definition talk:Neighborhood (Metric Space)

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There's an "expand" flag in this saying "Incorporate the standard $B(x; \epsilon)$ notation". But that's already there in the line:

"Open $\epsilon$-ball neighborhood of $a$ (and in deference to the word ball the notation $B_\epsilon \left({a}\right)$, $B \left({a, \epsilon}\right)$ or $B \left({a; \epsilon}\right)$ are often seen)"

Not sure quite what is meant. Am I missing something? --prime mover 02:01, 24 May 2012 (EDT)

't was late; never mind. I had a hunch that it was covered somewhere, but apparently hadn't gotten round to using the eyes. --Lord_Farin 03:11, 24 May 2012 (EDT)

Modern standard?

Is this really the standard modern usage of the word "neighborhood"? It doesn't coincide with Definition:Neighborhood (Topology). (Also, all the sources referenced were written before 1980.) --abcxyz (talk) 21:07, 11 October 2012 (UTC)

From the tone of your posting you obviously believe this not to be the case. The only works I possess were published before 1980. There are pages on this site somewhere which discuss different definitions of a neighborhood. If you have sources which offer a different definition, then feel free to share, as long as you do not amend the existing definition so as to be different from what it is, or there will be pages which rely on a definition which then does not match the premises of those pages and therefore will have been rendered unsound. --prime mover (talk) 22:26, 11 October 2012 (UTC)
For the record (I'm refactoring and tidying up --prime mover (talk) 23:05, 10 March 2014 (UTC)) I had this discussion after the above in a now-deleted redirected page:

Short of time today - I have a longish journey ahead of me - so this is going to be brief and rapidly expressed:

It seems the modern definition of "neighborhood" for a metric space is as defined here, and the old definition is as an open ball. The definition as given here for an open neighborhood is of course the same definition as used for Definition:Open Set (Metric Space).

The advantage to defining these concepts this way means that:

a): there exists the opportunity to talk about a non-open neighborhood, as this page does it
b): an open ball gets a page of its own as a separate concept from a neighborhood
c): the definition of "open neighborhood" in a metric space now corresponds with that of a topological space.

So the plan will be:

  1. Change the existing name of Definition:Neighborhood (Metric Space) to Definition:Open Ball (currently a redirect - there is no directly parallel concept in topology)
  2. Amend the current Definition:Neighborhood page from being a disambig to being a parent-and-transcluded-children page like with e.g. Definition:Continuity
  3. Change all the links to these pages and the language therein so as to change from "neighborhood" to "open ball" as appropriate.

It will just need some elementary refactoring (with my current mental workload I'm not intellectually prepared for anything more taxing).

Thoughts? I have to dash. --prime mover (talk) 06:16, 16 October 2012 (UTC)

Sounds good - go for it (when you have time). Be sure to separate e.g. open ball with just an Also see, that'd be more appropriate IMO. --Lord_Farin (talk) 06:35, 16 October 2012 (UTC)
Yep, no worries. I'll get to it - lots to catch up on today, everything I follow has gone wild. --prime mover (talk) 16:51, 16 October 2012 (UTC)

That page has now gone, and the discussion on this page ended up as abcxyz suggested.--prime mover (talk) 23:06, 10 March 2014 (UTC)