Talk:Neighbourhood system

Difference between a neighbourhood and a neighborhood
The following sentence invites us on a journey across oceans and cultures: "A neighbourhood of S in X is any subset T ⊆ X such that T contains some open neighborhood of S." Will someone qualified decide whether this page is to be written in British or American English? --Gabe — Preceding unsigned comment added by 135.180.2.43 (talk) 16:27, 7 January 2021 (UTC)

The union of local bases for all points x are a base for the topology
I don't see this being true. A neighbourhood of x (in the neighbourhood basis $$\mathcal{B}(x) $$ of x) may be closed - a base of the topology, however, only admits open subsets. Where does this come from? Quiet photon (talk) 08:39, 31 January 2011 (UTC)

Neighborhood filter undefined
The first Paragraph in Definitions mentions 'neighborhood filter' which is a link to this article which, as far as I can see, *nowhere* defines *what* a 'neighborhood filter' is. We are told that the neighborhood filter of a point is the same as the neighborhood filter of the singleton of that point, but not what the neighborhood filter *is*.

> Not an expert, but it appears that a 'neighborhood filter' is just another name for the neighborhood system (among others) which is simply calling attention to the fact that the neighborhood system happens to also be a filter. This is stated in the first line of the article. 65.128.22.54 (talk) 17:59, 19 February 2023 (UTC)


 * You are right. This is stated in the Neighbourhood filter subsection: The neighbourhood system for a point (or non-empty subset) is a filter called the neighbourhood filter for x.  It may be slightly confusing to have this terminology in the lead, but the reader can just take it at face value, given that the explanation comes later in the article.
 * As an aside, I think that the whole definitions section is a little too wordy and could be streamlined somewhat. Feel free to give it a shot. PatrickR2 (talk) 21:03, 19 February 2023 (UTC)

Too much duplication with the Neighbourhood article
It seems that about half of this article is a repeat of what's in the Neighbourhood article. This article should be trimmed down to what's specific about neighbourhood systems/filters/basis, and rely on links to the other article for the basics about neighbourhoods. PatrickR2 (talk) 05:00, 20 October 2022 (UTC)