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

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In summary, the conversation discusses the definition of a neighborhood of a point x and the confusion between two versions of it. The first version defines a neighborhood as an open set containing x, while the second version defines it as a set containing an open set of x. It is noted that these two versions are different and there is no right or wrong one. There is also a discussion about how this may cause confusion and how one version may be more general than the other. It is then mentioned that a neighborhood can be thought of as a collection of open balls and that there are many equivalent definitions.
  • #1
wayneckm
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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
 
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  • #2
A neighborhood of a point x is any set which contains the point x in its interior.
 
  • #3
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
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
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
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
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.
 

What is the definition of a neighborhood?

A neighborhood is a specific area within a larger community where people live in close proximity to one another and share common characteristics such as culture, socioeconomic status, and physical surroundings.

What are the key features of a neighborhood?

Some key features of a neighborhood include a shared sense of identity, a sense of community, and a common set of values and norms. Other features may include physical boundaries, a sense of safety and security, and access to local amenities and services.

How is a neighborhood different from a community?

While a neighborhood is a specific area within a larger community, a community is a broader term that encompasses a group of people who share common characteristics or interests. A community can include multiple neighborhoods within it.

What factors contribute to the formation of a neighborhood?

Some factors that contribute to the formation of a neighborhood include geographic location, historical development, social and economic factors, and the actions and interactions of its residents.

What are the benefits of living in a neighborhood?

Living in a neighborhood can provide a sense of belonging and connectedness, access to resources and support from neighbors, increased safety and security, and opportunities for social interaction and community involvement.

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