Difference between revisions of "Closure of a set in a topological space"

Latest revision as of 10:16, 28 September 2016

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^[1]:

• $\overline{A}:=\bigcap\{B\in\mathcal{P}(X)\ \vert\ A\subseteq B\wedge \underbrace{(X-B)\in\mathcal{J} }_{B\text{ is closed} } \}$ - 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

References

1. Jump up ↑ Introduction to Topological Manifolds - John M. Lee