Clopen set

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open and closed, and therefore clopen. As described by topologist James Munkres, unlike a door, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for doors is unrelated to their meaning for sets (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.

Examples

In any topological space

X

,

{\displaystyle X,}

the empty set and the whole space

X

{\displaystyle X}

are both clopen.

Now consider the space

X

{\displaystyle X}

which consists of the union of the two open intervals

(

0

,

1

)

{\displaystyle (0,1)}

and

(

2

,

3

)

{\displaystyle (2,3)}

of

R

.

{\displaystyle \mathbb {R} .}

The topology on

X

{\displaystyle X}

is inherited as the subspace topology from the ordinary topology on the real line

R

.

{\displaystyle \mathbb {R} .}

In

X

,

{\displaystyle X,}

the set

(

0

,

1

)

{\displaystyle (0,1)}

is clopen, as is the set

(

2

,

3

)

.

{\displaystyle (2,3).}

This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.