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There seems to be two common definitions of neighborhoods of a point- an open set containing the point, or an set containing an open set which itself contains that point. Why don't we just use the former definition, it seems so much simpler? What do we gain from the latter?

user6873235
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    Sometimes it is convenient that the set of neighborhoods of a point should be a "filter". This may fit in when you do a theory of convergence using filters. – GEdgar May 04 '18 at 10:40
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    @MichaelHoppe It is not a duplicate, in the question you mentioned the problem was to clarify Wikipedia's circular definition of open sets and neighbourhoods. The OP here asks what is the very point of having neighbourhoods that are not open. – Arnaud Mortier May 06 '18 at 22:13

4 Answers4

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Sometimes it's more convenient to have the more general definition : topologists talk about "closed neighbourhoods of a point", e.g. A space $X$ is regular iff every open neighbourhood of a point contains a closed neighbourhood of that point. Also, local compactness is often defined as "every point has a compact neighbourhood". These ways of defining the notions make it easy to have a general definition. It also nicely singles out the open sets: a set $A$ is open iff $A$ is a neighbourhood of $x$ for all $x \in A$.

Also, as @GEdgar notices in a comment, it's convenient to have the set of all neighbourhoods of a point be a so-called filter, and to make it closed under supersets we need the general definition too.

In short, it's not necessary, but it is convenient. And we can define things how we like, as long as it's clear what we mean.

Henno Brandsma
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It's a matter of convenience. Some times you need to talk about open sets containing the point, and some times you need to speak about sets containing an open set containing the point. So we need one name for each concept. We can (basically) choose between

  • Neighbourhood & set containing a neighbourhood
  • Open neighbourhood & neighbourhood

I personally think the second pair of names is more appealing, overall. So I would go for "neighbourhood of a point" meaning "a set which contains an open set which contains the point". Others may disagree, and they would think that "open set which contains the point" is a better definition.

Arthur
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The first definition makes it necessary that neighborhoods be open sets. The second definition allows closed sets to be neighborhoods as well. Many mathematicians require the neighborhood to be an open set. But it is a matter a convention. Hence we have two definitions.

Sonal_sqrt
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    I'm not sure how this answers the question. The question is "Why do we use definition A instead of definition B? What's the advantage of A?" and your answer is "Some people use A and some people use B", which seems to address neither part. – David Richerby May 04 '18 at 12:45
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In German we have two definitions. An “Umgebung” of $x$ is any open set that contains $x$. A “Nachbarschaft” of $x$ is any sets that contains an Umgebung of $x$. I didn’t find a proper translation of those two words.

Michael Hoppe
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  • In Horst Herrlich (Topologie I): 1.2.7: "$B$ heißt Umgebung von $A$ in $X$, falls eine offene Menge $C$ in $X$ mit $A \subset C \subset B$ existiert. Ist $B$ Umgebung von ${x}$ so heißt $B$ auch Umgebung von $x$, und $x$ heißt innere Punkt von $B$." So it's not generally used as such in German it seems. What book do you use? – Henno Brandsma May 04 '18 at 11:29
  • von Querenburg, "Mengentheoretische Topologie" has the same definition of "Umgebung" as Herrlich. Both do not mention "Nachbarschaft". Dutch only uses "omgeving", Frisian "omjouwing", both cognates of "Umgebung". – Henno Brandsma May 04 '18 at 11:34
  • Those definitions came from lectures of Peter Bombrowski. – Michael Hoppe May 04 '18 at 16:22
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    Then it’s probably his own choice of words. – Henno Brandsma May 04 '18 at 16:37