Closed Set

There are several equivalent definitions of a closed Set. A Set $S$ is closed if

1. The Complement of $S$ is an Open Set,

2. $S$ is its own Closure,

3. Sequences/nets/filters in $S$ which converge do so within $S$,

4. Every point outside $S$ has a Neighborhood disjoint from $S$.

The Point-Set Topological definition of a closed set is a set which contains all of its Limit Points. Therefore, a closed set $C$ is one for which, whatever point $x$ is picked outside of $C$, $x$ can always be isolated in some Open Set which doesn't touch $C$.

See also Closed Interval