Open and Closed Sets

In Metric Space

Let be a metric space and let be a subset of

  • is closed if it contains all of its boundary points, i.e., .
  • is open if it contains none of its boundary points, i.e., .
  • If contains some of its boundary points but not others, it is neither open nor closed.

For example, in the real line with metric , is open, is closed, and is neither open nor closed.

It is possible for a set to be simultaneously open and closed, if it has no boundary (when every point in is an interior point). Under many metrics, the empty set is the only such example, but there are exceptions.

Notice that the definition of open and closed set depends on the choice of metric. For instance, on with metric , is not open. However, if we uses the discrete metric , then is now an open set.