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I believe topology was defined in order to talk about limit points. In order to say things like 'point x is a limit point of set E' or, equivalently, 'x can be arbitrarily approximated by points in E'. The realization was that we don't actually need a metric. All we need is a notion of 'neighborhood', i.e. a little in every direction of a point.

With the notion of neighborhood, we can now define limit point, interior, exterior, boundary, open set, closed set, etc. For example, a boundary point of a set E is a point x such that every neighborhood that contains x contains a point in E and a point not in E. An open set is a set those points are all interior points, and so on.

All this seems intuitive. It makes sense that neighborhoods would be closed under arbitrary union and finite intersection, for example. What I do not understand is why the neighborhoods are assumed to be the open sets in a topology. Is there any reason behind this?

fdzsfhaS
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  • I don't understand your question...if, as you say, neighborhoods are closed under arbitrary unions and finite intersections, then they are the open sets of a topology, by definition. – Eric Wofsey Jun 09 '18 at 20:22
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    There is a different notion of "neighborhood" where one speaks only of a neighborhood of a point $x$ and such sets are not the same as open sets (a neighborhood of $x$ is a set that contains an open set containing $x$), but that doesn't seem to be what you are talking about. – Eric Wofsey Jun 09 '18 at 20:24
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    I must misunderstand your question. To me it reads: 'Why do we call the members of a topology open sets?' And that's, at least to me, just a naming convention. – Stefan Mesken Jun 09 '18 at 20:26
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    I think I'd would be a good idea to state the definition of neighborhood that you are working with, and where it is stated that neighbourhoods must be open. – Jendrik Stelzner Jun 09 '18 at 20:31
  • Stefan Mesken: open sets are sets whose points are interior points. That notion predates the definition of a topology. If I am correct in assuming that when defining a topology, they were trying to formalize the intuition of a neighborhood, then it doesn't seem natural to make the neighborhoods and the open sets the same as is standardly done. – fdzsfhaS Jun 09 '18 at 20:32
  • Jendrik Stelzner: Neighborhoods are the intuition of around a point that I believe the structure of a topology was trying to formalize – fdzsfhaS Jun 09 '18 at 20:35
  • Eric Wofsey: I'm trying to see if there is indeed a reason why the neighborhoods are defined to be the open sets. – fdzsfhaS Jun 09 '18 at 20:36
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    Typically - at least hereabouts - a neighbourhood is not assumed to be open. Some people define neighbourhoods to be open, others don't. Both conventions make the phrasing of things easier in some places and more cumbersome in others. – Daniel Fischer Jun 09 '18 at 20:42
  • Why do you think "it doesn't seem natural to make the neighborhoods and the open sets the same"? You haven't given any explanation for this. – Eric Wofsey Jun 09 '18 at 21:45
  • If your intuitive definition of "neighborhood" is "a set that contains a little in every direction of each of its points", why wouldn't that be the same a set whose points are all interior points? – Eric Wofsey Jun 09 '18 at 21:51
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    See here why it's convenient to have neighbourhoods be a more general notion than openness. – Henno Brandsma Jun 10 '18 at 07:55

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I think a neighbourhood space is what was originally used to formalise the idea of a neighbourhood. Here every point has a non-empty set of neighbourhoods $\mathcal{N}(x)$, such that

  1. Every $x$ is in every one of its neighbourhoods, or formally: $$\forall x : \forall N \in \mathcal{N}(x): x \in N$$

  2. Neighbourhoods are closed under intersections: $$\forall x \forall N_1, N_2 \in \mathcal{N}(x): N_1 \cap N_2 \in \mathcal{N}(x)$$

  3. Neighbourhoods are closed under enlargements: $$\forall x \forall N \in \mathcal{N}(x): \forall N' \subseteq X: (N \subseteq N') \to N' \in\mathcal{N}(x)$$

Note that 2 and 3 can be reformulated as saying $\mathcal{N}(x)$ is a filter on $X$ and 1 that is refined by the fixed filter on $x$. Also note that we only talk about "neighbourhood of $x$" not just "neighbourhood".

This set-up allows one to define interior point: $x$ is an interior point of $A$ iff $A \in \mathcal{N}(x)$ and from this the interior of a set $A$, the closure of $A$ etc., etc. A set is called open if it's its own interior, or $$\forall x \in O: O \in \mathcal{N}(x)$$ and this defines a topology on $X$. An open set is a neighbourhood of each of its points.

In this topology, being a neighbourhood of $x$ does not mean that that set is always open. E.g. $[-1,1]$ is a neighbourhood of $0$ in the usual topology but not open. If we take the set of all neighbourhoods of all points together (their union) then this is not closed under intersections: $[-1,1] \in \mathcal{N}(0)$, $[1,3] \in \mathcal{N}(2)$ but $\{1\} = [-1,1] \cap [1,3]$ is not a neighbourhood of any point, again in that topology. Open sets are special, because they are neighbourhoods of all its points.

Neighbourhoods in most texts are not assumed to be members of a topology, just that $N$ is a neighbourhood of $x$ if there is some open set $O$ just that $x \in O \subseteq N$. Open sets are neighbourhoods, but not reversely. I talked about why that can be convenient here.

Henno Brandsma
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  • Actually, in topology, being a neighborhood of $x$ DOES mean it is open. I.e. a neighborhood of $x$ is defined to be an open set $U \ni x$. At least in Munkres... – Sigurd Jun 10 '18 at 09:05
  • @Sigurd Actually, in topology, a set $N$ is a neighborhood of a point $x$ if there is an open set $U$ such that $x\in U\subseteq N.$ That's how neighborhood is defined in my topology books. Who is this Munkres person? – bof Jun 10 '18 at 09:09
  • https://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292 – Sigurd Jun 10 '18 at 09:11
  • Actually I find your definition to make more sense. I remember I was confused about the fact that a neighborhood is always open when I read the Munkres definition. – Sigurd Jun 10 '18 at 09:13
  • Which books do you use? – Sigurd Jun 10 '18 at 09:14
  • @Sigurd Kelley and Willard. (But I haven't cracked either one for a long time, so I'm half expecting someone to tell me that I'm wrong and they use the same bad definition of neighborhood that this Munkres person does.) – bof Jun 10 '18 at 09:29
  • Yeah I guess it's just a matter of convention. – Sigurd Jun 10 '18 at 09:39
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    @Sigurd Munkres defines (in edition 1, p96 ) "Mathematicians often use some special terminology here. They shorten the statement '$U$ is open set containing $x$' to the phrase '$U$ is a neighborhood of $x$'" and later on that page "Some mathematicians use the term "neighborhood" differently. They say that $A$ is a neighborhood of $x$ if $A$ merely contains an open set containing $x$. We shall not follow that practice". – Henno Brandsma Jun 10 '18 at 09:47