# What is a neighborhood?

• Apr 27th 2010, 11:33 AM
lrl4565
What is a neighborhood?
Is it:

a) a set A that has a subset B with a point P in it (in which case, the point P's neighborhood is set A)

b) "Set V is a neighborhood of point P if a small disk around P is contained in V" - Wikipedia. What is this small disk? Is it a set? Is there an established radius?
• Apr 27th 2010, 11:48 AM
Moo
Quote:

Originally Posted by lrl4565
Is it:

a) a set A that has a subset B with a point P in it (in which case, the point P's neighborhood is set A)

No, B has to be open. And in which case, A is one of P's neighbourhoods.

Quote:

b) "Set V is a neighborhood of point P if a small disk around P is contained in V" - Wikipedia. What is this small disk? Is it a set? Is there an established radius?
No established radius. This definition only holds for metric spaces.
It's more commonly called a ball
By the way, in Wikipedia, they talk about a ball, not a disk...
• Apr 27th 2010, 11:55 AM
Drexel28
[QUOTE=lrl4565;502367]Is it:

a) a set A that has a subset B with a point P in it (in which case, the point P's neighborhood is set A)

Quote:

Originally Posted by Moo
No, B has to be open. And in which case, A is one of P's neighbourhoods.

I agree that \$\displaystyle B\$ needs to be open, but there is some dispute between the exact definition of a neighborhood. Some French mathematicians following Bourbaki sometimes define a neighborhood of a point to be any set containing an open set containing that point, whereas more often a neighborhood is just an open set containing the point.
• Apr 27th 2010, 12:09 PM
Moo
Quote:

Originally Posted by Drexel28

I agree that \$\displaystyle B\$ needs to be open, but there is some dispute between the exact definition of a neighborhood. Some French mathematicians following Bourbaki sometimes define a neighborhood of a point to be any set containing an open set containing that point, whereas more often a neighborhood is just an open set containing the point.

That's what your Wikipedia says... I also read the Spanish Wikipedia, which states the same. So I'd like to know where your red "more often" comes from !!

Maybe you're confusing with the fact that an open set is a neighbourhood of all its points.
• Apr 27th 2010, 12:10 PM
Moo
Quote:

Originally Posted by Drexel28

I agree that \$\displaystyle B\$ needs to be open, but there is some dispute between the exact definition of a neighborhood. Some French mathematicians following Bourbaki sometimes define a neighborhood of a point to be any set containing an open set containing that point, whereas more often a neighborhood is just an open set containing the point.

That's what your Wikipedia says... I also read the Spanish, German & Italian Wikipedia, and they write the same definition. So I'd like to know where your red "more often" comes from !!

Maybe you're confusing with the fact that an open set is a neighbourhood for all of its points.
• Apr 27th 2010, 12:12 PM
Drexel28
Quote:

Originally Posted by Moo
That's what your Wikipedia says... I also read the Spanish Wikipedia, which states the same. So I'd like to know where your red "more often" comes from !!

I'm not saying it's the world standard, I'm saying that in most of the 32 topology books I own that is how a neighborhood is defined.

Quote:

Maybe you're confusing with the fact that an open set is a neighbourhood of all its points.
I hate this expression, it personally aways confuses me. This is the approach taken by Bert Mendelson in his "Introduction to Topology" and otherwise good book.
• Apr 27th 2010, 12:14 PM
Moo
Defining it this way is more laziness than something else :D
You're just skipping the 'useless' part of the definition !

So tell me, out of 32 of these books, how many define it in the second way ?
And out of these 32 books, how many are written by American authors ?
American school is not alone in the world :P

I'm sorry I'm afraid I can't decrypt Russian, so I don't know what the 'Russian standard' is, according to Wikipedia.

Quote:

I hate this expression, it personally aways confuses me.
It's a property that is always true. Even if it confuses you, it won't prevent people from using it.
• Apr 27th 2010, 12:17 PM
Drexel28
Quote:

Originally Posted by Moo

So tell me, out of 32 of these books, how many define it in the second way ?

3:

General Topology- Bourbaki Parts I and II

Introduction to Topology-Bert Mendelson

And my favorite professor

Topological Methods in Euclidean Space-Greg Naber

Quote:

And out of these 32 books, how many are written by American authors ?
Most

Quote:

American school is not alone in the world :P
Thus me saying it's not a world standard. Plus, the OP is from America (supposedly)
• Apr 27th 2010, 12:18 PM
Moo
Quote:

Thus me saying it's not a world standard. Plus, the OP is from America (supposedly)
And so what ? I'm correcting the definition he wrote. And the way he wrote it, it is more likely the first than the second one !!!
Slow it down, create a new thread, it's not the topic here !
• Apr 27th 2010, 01:02 PM
lrl4565
Alright, what I've gathered:

Hypothetical situation:

Set A: [1,10]
Set B: (3, 7)
Point P: 6

Both Set A and Set B are neighborhoods of point P
• Apr 27th 2010, 02:07 PM
Drexel28
Quote:

Originally Posted by lrl4565
Alright, what I've gathered:

Hypothetical situation:

Set A: [1,10]
Set B: (3, 7)
Point P: 6

Both Set A and Set B are neighborhoods of point P

That depends on your definition once again and what you call "open".

If I were a teacher and I was teaching you about neighborhoods with the usual real number structure than I would say B is a neighborhood but A isn't.
• Apr 27th 2010, 10:05 PM
lrl4565
So, there's dispute over the exact definition of a neighborhood. I am American. What is the most common?

What IS a neighborhood? Why categorize subset B (with point P inside) of set A as the neighborhood of P? Labeling purposes, or does it mean something more?
• Apr 27th 2010, 10:24 PM