Because I read different book find there are two definition and they aren't look like a same thing.
Definition 1 from GTM27
A set is open if it's a member of a topology,
and topology is a family of sets which satisfies:
1)the intersection of any two members is still a member.
2)the union of any two members is still a member
Definition 2 from baby rudin
Let X be a metric space. All points and sets mentioned below are understood to be elements and subsets of X.
A set S is open if every point of S is an interior point of S.
and a point p is an interior point of X if there is a neighborhood N of p such that N is contained by X.
A neighborhood of a point p is a set N(P) consisting of all points q such that d(p,q)< r.
definition 2 is complex.
So I try to make it become simpler to find the relation between the two concept.
I thought a set S is open if for every point p of S there is a set N such that N is the subset of X and consist of all points q that d(p,q)
But I still don't find any similarities between the two concept.
In first definition,we know the definition of open set only depends on the concept of set.
But the second definition tell us the definition depends on point and the set.
This is what I am confused now.