Closed Set: Definition & Example

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you'll see how both the theoretical definition of a closed set and its real world application. You'll learn about the defining characteristic of closed sets and you'll see some examples.

A Closed Set

Math has a way of explaining a lot of things. And one of those explanations is called a closed set. In math, its definition is that it is a complement of an open set. This definition probably doesn't help. So, you can look at it in a different way. You can think of a closed set as a set that has its own prescribed limits. An open set, on the other hand, doesn't have a limit. If you include all the numbers that you know about, then that's an open set as you can keep going and going. Your numbers don't stop.

A Closed Set Has a Boundary

But, if you think of just the numbers from 0 to 9, then that's a closed set. It has its own prescribed limit. It has a boundary. If you look at a combination lock for example, each wheel only has the digit 0 to 9. You can't choose any other number from those wheels. Each wheel is a closed set because you can't go outside its boundary.

You can also picture a closed set with the help of a fence. Look at this fence here. It's a round fence. People can exercise their horses in there or have a party inside.

closed set

Now, which part do you think would make up your closed set? Is it the inside of the fence or the outside? If you picked the inside, then you are absolutely correct! The inside of the fence represents your closed set as you can only choose the things inside the fence.

closed set

The outside of the fence represents an open set as you can choose anything that is outside the fence.

closed set


A closed set is a different thing than closure. When a set has closure, it means that when you perform a certain operation such as addition with items inside the set, you'll always get an answer inside the same set. A set that has closure is not always a closed set. For example, the set of real numbers, for example, has closure when it comes to addition since adding any two real numbers will always give you another real number. However, the set of real numbers is not a closed set as the real numbers can go on to infinity. The set is not completely bounded with a boundary or limit.

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