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

Given a Topological Space (S,T), a setN[tex]\subset[/tex] S is a neighborhood of a pointx[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]NandN[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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# Definition of Neighborhood, Very Confusing

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