# 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.[1]

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.[2]

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 ${\displaystyle \left[{\frac {h}{2}},{\frac {3h}{2}}\right]}$, 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.

## References

1. "Open Set". Wolfram MathWorld.
2. "Closed Set". Wolfram MathWorld.