Open set

Example: The blue circle represents the set of points (x, y) satisfying x² + y² = r². The red disk represents the set of points (x, y) satisfying x² + y² < r². The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.

In set theory an open set is a set where all elements have the same properties. Simply put, an open set is a set that does not include its edges or endpoints. For each point in the set, you can make a bubble around that point, such that all points in the bubble are also in the set.

On the other hand, a closed set includes all its edges or endpoints. A set that includes some of its edges or endpoints is neither open nor closed.

An open set is very similar to an open interval.

Examples

The set (0,1) is open. If we choose a very small value, there will always be a small bubble which are all in the set (0,1).

If we choose a very small value h ∈ (0,1), we can make a bubble [math]\displaystyle{ \left [ \frac{h}{2},\frac{3h}{2} \right ] }[/math], in which all the values are in (0,1).

However, [0,1] is closed. If we choose the value 0, and choose a very small value k, 0-k ∉ [0,1], which means that it's closed.

Open Set Media

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  Example: the blue circle represents the set of points (x, y) satisfying x2 + y2 = r2. The red disk represents the set of points (x, y) satisfying x2 + y2 < r2. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.