Definition of neighborhood

1. Jul 14, 2010

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

2. Jul 14, 2010

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

3. Jul 14, 2010

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)".

4. Jul 14, 2010

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.

5. Jul 14, 2010

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".

6. Jul 14, 2010

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.

7. Jul 18, 2010

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.