Thread: metric spaces

Let (M,d) be a metric space.
1. Choose x element of M. Show that a set U subset of M is a neighbourhood of x if and only if x is an element of the Interior of U
2. Prove that the intersection U intersection V of any two neighbourhoods U and V of a point x element of M is also a neighbourhood of x.

Originally Posted by Godisgood

Let (M,d) be a metric space.
1. Choose x element of M. Show that a set U subset of M is a neighbourhood of x if and only if x is an element of the Interior of U
2. Prove that the intersection U intersection V of any two neighbourhoods U and V of a point x element of M is also a neighbourhood of x.

Thanks soo much. I am really struggling with this since it is the first time I am taking a course of this nature, so please go in steps
Thanks for the help

What is a neighborhood? Do you take the Bourbaki "set containing an open set containing the point" or just "open set containing point"

Originally Posted by Drexel28

What is a neighborhood? Do you take the Bourbaki "set containing an open set containing the point" or just "open set containing point"

@Godisgood,
Please answer the above question.

set containing an open set containing the point

O.K. then if is an open set and the by definition , interior of .
On the other,hand, if the by definition of interior, an open set such that so is a neighborhood of .

If each of is a neighborhood of there are open sets .
But is open and .