What is the difference between the two definitions of a neighborhood?

[wayneckm]

Hello all,


Indeed I am quite confused with the definition of neighborhood of a point x since I come across two versios of it.

The first one is simply defined as open set of x while the second one is defined as a set containing an open set of x.

Apparently these two are different notions, e.g. (x-a, x+a], in first one this is not a neighborhood while the second one it is.

So which one is the right one? Thanks very much.


Wayne

 

[radou]

A neighborhood of a point x is any set which contains the point x in its interior.

 

[wayneckm]

So are you saying the second version is the right one?

I could read in Munkres's book that he said "we shall avoid (the second version of neighbourhood)".

 

[quasar987]

Yes, well it is just a convention. Given a text, you just have to figure out which convention they are using. There is no right or wrong one.

 

[wayneckm]

But won't this cause confusions and troubles when one proves with the use of neighborhood? I mean somehow the theorems may depend on the particular choice of "neighborhood"?

Apparently the second convention is more general since in some sense we can "give name to more sets".

 

[quasar987]

you are afraid that some thm statement involving one version of "nbhd" might not be true when u replace the meaning of "nbhd" with the other version?

That's true, but there is always an analogous statement, since a nbhd in version 1 is also a nbhd in version 2 and every nbdh of version 2 contains a nbhd of version 1.

 

[Cryxic]

You can think of a neighborhood as a collection of open balls, which is equivalent to many other definitions. For example, with regard to the two definitions you gave, a set containing an open set of x can be that open set of x itself. Remember, all sets contain themselves, by definition! In general, however, an open ball will be a proper subset of some neighborhood, which itself will be some open proper subset of the original set.