exterior product

In

topology

, the exterior of a subset $S$ of a topological space $X$ is the union of all open sets of $X$ which are

disjoint

from $S.$ It is itself an open set and is disjoint from $S.$ The exterior of $S$ in $X$ is often denoted by $\operatorname_X S$ or, if $X$ is clear from context, then possibly also by $\operatorname S$ or $S^.$

Equivalent definitions

The exterior is equal to $X \setminus \operatorname_X S,$ the complement of the (topological) closure of $S$ and to the (topological) interior of the complement of $S$ in $X.$

Properties

The topological exterior of a subset $S \subseteq X$ always satisfies:
:$\operatorname_X S = \operatorname_X (X \setminus S)$

and as a consequence, many properties of $\operatorname_X S$ can be readily deduced directly from those of the interior $\operatorname_X S$ and elementary set identities. Such properties include the following:

* $\operatorname_X S$ is an open subset of $X$ that is disjoint from $S.$
* If $S \subseteq T$ then $\operatorname_X T \subseteq \operatorname_X S.$
* $\operatorname_X S$ is equal to the union of all open subsets of $X$ that are disjoint from $S.$
* $\operatorname_X S$ is equal to the largest open subset of $X$ that is disjoint from $S.$

Unlike the interior operator, $\operatorname_X$ is not idempotent, although it does have the following property:

* $\operatorname_X S \subseteq \operatorname_X \left(\operatorname_X S\right).$

See also

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Bibliography

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{{Authority control
General topology