Closure (topology)

In topology and the Analysis the closed hull (also seclusion or conclusion ) is a subset of a topological or metric space, the smallest closed superset of.

Definition

If a topological space, then the closed hull or the completion of a subset of the intersection of all closed subsets of that contain. The amount is even finished, so it is the smallest closed superset of.

One point of contact means or Adhärenzpunkt of if every neighborhood of at least one element of contain. consists exactly of the contact points of.

The financial statements as a set of limits

Complete the first axiom of countability (this applies, for example, if a metric space is ), then the set of all limits of convergent sequences whose terms are in.

Is an arbitrary topological space, the completion of a subset is the set of limits of convergent networks, whose members are in.

Completion of balls in metric spaces

It is a metric space with metric. It should be noted that in general the open closure of a ball

With radius and center is not the same as the closed ball

Since the closed ball is a closed set that contains the open ball, it also includes its financial statements:

To give an example in which this inclusion is strict, let X be a set ( with at least two elements ) on which a metric by

Is defined. Then for each:

In addition, there are also metric spaces in which a radius r both inclusions are simultaneously true for a point x and:

One example is the amount with the induced by Euclidean space metric. Here met the specified inclusion condition: