# Definition of Neighborhood, Very Confusing

1. Aug 4, 2010

### 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

2. Aug 4, 2010

### quasar987

The same question popped up not long ago: