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**Plato** Be very careful with statements such as that.

A door is either open or closed. That ain't true for sets.

The set $\displaystyle [0,1)$ is neither open nor closed.

Its complement $\displaystyle (-\infty,0)\cup [1,\infty)$ is also neither open nor closed

If is true that a set is closed in and only if its complement is open.

As has already been pointed out, the standard metric space definition of closed sets is:

**A set is closed if and only if it contains all of its limit points.**