Contents

Idea

Georg Cantor used to define the continuum as a perfect space that is connected as well. Hence, the property of ‘being perfect’, as the name indicates as well, can be viewed as forming part of the concept of a ‘prototypical’ topological space.

Definition

A topological space $X$ is perfect if it has no isolated points, i.e., if every point $x$ belongs to the topological closure of its complement $X \setminus \{x\}$.

In a topological space $X$, a subset is said to be perfect if it is closed in $X$ and perfect in its subspace topology.

Properties

• Every topological space $X$ is the disjoint union of a scattered subset and a perfect subset.

Reference

• S. Willard, General Topology , Addison-Wesley Reading 1970. (Dover reprint 2004, pp.216ff)

Last revised on May 9, 2017 at 02:59:12. See the history of this page for a list of all contributions to it.