Definition of Neighborhood, Very Confusing

In summary, there are two different definitions of a neighborhood of a point, one stating that it is any set that contains an open set which contains the point, and the other stating that it is an open set that contains the point. These definitions are not equivalent and it is important to pay attention to which definition is being used when reading texts on the subject.
  • #1
jetplan
15
0
Hi All math lovers,

I have seen 2 different definition of a neighborhood of a point. Which one is correct ?

Given a Topological Space (S,T), a set N [tex]\subset[/tex] S is a neighborhood of a point x [tex]\in[/tex] S iff

1. [tex]\exists[/tex] U [tex]\in[/tex] T, such that x [tex]\in[/tex] U [tex]\subseteq[/tex] N

i.e. a neighborhood of a point is any set that contains an open set which in turns contains that point. The neighborhood itself need not be open.

OR

2. x [tex]\in[/tex] N and N [tex]\in[/tex] T

i.e. a neighborhood of a point is any OPEN set that contains that point. Therefore, a neighborhood must be open.

REAL ANALYSIS and PROBABILITY by RM DUdley suggests (1)
TOPOLOGY by James Munkres suggest (2)

I am really confused by this. Anyone shed some light ?

Thank you so much
 
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  • #2
The same question popped up not long ago:

https://www.physicsforums.com/showthread.php?t=415925

The short answer is that you're right in being confused: the two definitions are not equivalent, and you have to be careful when you read a text to determine which definition the author is using.
 
  • #3
Sometimes definition 2 is called an open neighbourhood, to avoid confusion.
 

Related to Definition of Neighborhood, Very Confusing

What is the definition of a neighborhood?

A neighborhood is a specific area or community within a larger city or town where people live, work, and interact with each other.

What makes a neighborhood confusing?

A neighborhood can be confusing due to its size, layout, or the diverse mix of people and cultures that reside within it.

How is a neighborhood defined by researchers?

Researchers often define a neighborhood based on geographical boundaries, such as streets, zip codes, or census tracts, and also take into account social and economic factors like income levels, education, and crime rates.

What are some common characteristics of a neighborhood?

A neighborhood typically has a sense of community and shared identity, local businesses and amenities, and a mix of residential and commercial properties.

Can a neighborhood change over time?

Yes, a neighborhood can change over time due to factors such as urban development, demographic shifts, and economic changes. This can lead to the gentrification or decline of a neighborhood.

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