Closed Set vs. Open Set

Instructor: Yuanxin (Amy) Yang Alcocer

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

In math, there's two different types of sets. These sets are there to help you with numerical boundaries and limits. They help you determine whether a number you are looking at is part of a set or not. Learn how in this lesson.


Did you know that every time you work with a math problem, you are working with a set of some kind? A set is a collection of items. In math, your sets are usually made up of numbers that share some characteristic.

When you first started learning about math, you learned about your counting numbers. You began with counting your fingers and toes. Then as you continued learning about math, you then learned about negative numbers, even numbers, odd numbers, and so forth. All of these are sets.

Sometimes, though, you'll have math problems with limits in them. When you have this situation, your sets get separated into two different types, closed sets and open sets.

Open Sets

First, let's talk about open set. Put simply, an open set is a collection of numbers that does not include any limit points. Think of it as an interior point that never touches the boundary or wall, so to speak. In math theory speak, an open set includes all the points inside the set such that any point can have a bubble or ball around it without touching another point.

This may sound complicated, but it's really not. Just think of it as an open bubble on your number. Remember back to elementary school, when you would draw points and answers on the number line. If you had an open bubble, it meant that the particular point in question was not included in the answer, but it was a limit. For example, when you had x < 3, you would draw an open bubble on the 3 and then draw a thick line going towards the left:

open set closed set

And there you have an open set. This doesn't mean, though, that your open sets are infinite. No, your open sets can also be bounded in more than one direction, such as this:

  • 0 < x < 3

You have an open ball on either side and your open set includes all the numbers between 0 and 3. Your numbers never touch the 0 and 3, thus all your points are interior or inside points:

open set closed set

Now, because the numbers in your set never touch the numbers of 0 and 3, you can have a bubble, even if it's a very small one, around each and every point in your set. You can get as close as you want to your limits of 0 and 3 and you can still create a very tiny bubble around your point. It still won't touch your boundary. Therefore, 0 < x < 3 is also an open set.

Closed Set

Now, let's talk about closed sets. Mathematically, the definition of a closed set is the complement of an open set. Another way to define this is to say that a closed set contains the boundaries or limits of the set.

On the number line, it means you have a solid ball or bubble instead of an open one. One way to determine if you have a closed set is to actually find the open set. The closed set then includes all the numbers that are not included in the open set.

For example, for the open set x < 3, the closed set is x >= 3. This closed set includes the limit or boundary of 3. For the open set of 0 < x < 3, the closed set is x <= 0 and x >= 3. All of these include the boundary or wall from the open set.

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