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Definition of Neighborhood, Very Confusing

  1. Aug 4, 2010 #1
    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
     
  2. jcsd
  3. Aug 4, 2010 #2

    quasar987

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    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.
     
  4. Aug 4, 2010 #3
    Sometimes definition 2 is called an open neighbourhood, to avoid confusion.
     
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