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

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In summary, the notion of closure refers to a set being closed under a specific operation or topology. This means that the set either remains unchanged or its closure is a subset of another set. This can be seen in various definitions of closedness, such as a set being closed if it contains all its accumulation points or if its complement in another set is open. It is important to note that this should not be confused with the concept of closure in topology.
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jack1234
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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?
 
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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)
 
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jack1234 said:
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.
 

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

What is the notion of closure?

The notion of closure refers to the idea that a set of mathematical operations or functions will always result in a value within the same set. It is a property that ensures that the result of an operation on elements within a set will also be within the same set.

Why is the notion of closure important in mathematics?

The notion of closure is important in mathematics because it allows for the creation of new mathematical structures and the ability to perform operations on those structures without leaving the set. It also helps to define and prove properties of these structures.

How is the notion of closure related to algebraic structures?

The notion of closure is a fundamental property of algebraic structures such as groups, rings, and fields. These structures are defined by a set of elements and operations that are closed under those operations, meaning that the result of any operation will always be an element of the same structure.

Can you give an example of the notion of closure in action?

One example of the notion of closure is in the set of real numbers under addition. If we take any two real numbers and add them together, the result will always be another real number. This demonstrates the closure property of the real numbers under addition.

What happens if a set does not have closure under a certain operation?

If a set does not have closure under a certain operation, it means that performing that operation on elements within the set may result in a value that is not within the set. This can lead to unexpected or undefined behavior and can limit the use and applicability of that set in mathematical operations.

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