What Does It Mean When a Set Is Closed Under Another Set?

[jack1234]

What is "notion of closure"

May I know what is "notion of closure", I don't need comprehensive explanation, I just want to understand what is meant by a set is closed under another set in order to proceed.

For example, Set B is closed under Set A,
does it mean that Set A is a strict superset of Set B?

 

[quasar987]

Possible definitions of closedness* are

1) B is closed in A if "All sequence of elements of B converge to an element of B". In other words, "B is closed, if it contains all its accumulation points".

2) By boundary of B, we mean the accumulation points of B for which any open ball centered on them dips in the complement of B in A. B is closed if it is the union of its interior with its boundary. (The union of the interior with the boundary is called the closure of B, such that the closedness of B can be defined with more ease by saying that B is closed if it equals its closure.)

3) A set B is closed in A if it's complement in A is open.

I was going from memory. You can get the real definitions there: http://en.wikipedia.org/wiki/Closed_(mathematics)
http://mathworld.wolfram.com/ClosedSet.html

*not to be confused with closure (http://en.wikipedia.org/wiki/Closure_(topology)#Closure_of_a_set)

 

[HallsofIvy]

May I know what is "notion of closure", I don't need comprehensive explanation, I just want to understand what is meant by a set is closed under another set in order to proceed.

For example, Set B is closed under Set A,
does it mean that Set A is a strict superset of Set B?

I've never heard of a set being "closed under" another set.

I've heard of a set being "closed under" a specific operation- that means that when you apply the operation to one or more members of the set the result is still in the set.

I've heard of set being "closed" in some topology but that has nothing to do with "under set A". (That's what quasar987 is talking about.)

My best guess might be that, assuming A and B are sets in some topological space, then not only B but the closure of B is a subset of A.