Clopen set

In subdivision topology of mathematics is a closed open set (set clopen in English, rarely in German also abgeschloffene amount ) a subset of a topological space that is simultaneously completed and open.

This seems odd at first glance; but you have to remember that the terms " open" and "closed" in the topology have a different meaning than in everyday speech.

Examples

In any topological space is the empty set and the whole space are completed and open. In a related topological space these are the only subsets that are closed and open.

In the topological space X, which consists of the union of the two intervals and, provided with the topology induced from the standard topology, the set is closed and open.

In general, a connected component of an area is not open and closed: In the Alexandroff compactification of the set of the natural numbers, the point at infinity forms a connected component that is not open.

Consider the set of rational numbers with the standard topology, and in the subset A of all rational numbers greater than (or equivalently here: at least as large as ) are the square root of 2. Since is irrational, one can easily show that A is closed and open. Note, however, that A is not completed as a subset of the real numbers still open; the set of all real numbers is greater than open but not complete, while the set of all real numbers that are at least as large as completed but is not open.

Properties

• A subset of a topological space if and only open and closed when its edge is empty.
• A topological space X is then connected if the only closed open sets are the empty set and X.
• Each completed open subset can be represented as a ( possibly infinite ) union of connected components.
• If each connected component is open (which for example is the case when X has only finitely many components, or if X is locally connected ), then any union of connected components is complete and open.
• A topological space is discrete then if every subset is open and closed.
• For every topological space which concluded open sets form a Boolean algebra.
• An open subset of a topological group is also closed. A closed subgroup of finite index is also open.