- #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
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