Definition of Neighborhood, Very Confusing

[jetplan]

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 \subset S is a neighborhood of a point x \in S iff

1. \exists U \in T, such that x \in U \subseteq 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 \in N and N \in 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

 

[quasar987]

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.

 

[adriank]

Sometimes definition 2 is called an open neighbourhood, to avoid confusion.