Closure of a set in a topological space

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

There is an (ancient) page, Closure, interior and boundary, ensure all information there is present here and then turn that into an overview page. This page is marked A* in grade because of the importance of the closure, interior and boundary concepts

Note: closure is an important term in mathematics (eg a group is "closed" under its operation), hence the specific name. This name must be inline with the closely related concepts of interior of a set in a topological space and boundary of a set in a topological space, boundary is the reason closure (topology) couldn't be used as even in topology "boundary" has several meanings.

• closure (set, topology) redirects here, for use with template:link.

Definition

Let $(X,\mathcal{ J })$ be a topological space, let $A\in\mathcal{P}(X)$ be an arbitrary subset of $X$. The closure of $A$, denoted $\overline{A}$, is defined as follows:

• $\overline{A}:=\bigcap\{B\in\mathcal{P}(X)\ \vert\ A\subseteq B\wedge \underbrace{(X-B)\in\mathcal{J} }_{B\text{ is closed} } \}$ - the intersection of all closed sets which contain $A$
  • Recall, by definition, that a set is closed if its complement is open and that $X-B$ is another way of writing the complement of $B$ in $X$

See also

• Closure, interior and boundary
  • Interior of a topological space
  • Boundary of a topological space