Open and Closed Sets
In Metric Space
Let
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
It is possible for a set to be simultaneously open and closed, if it has no boundary (when every point in
Notice that the definition of open and closed set depends on the choice of metric.
For instance, on