Kompaktifikáció

Kompaktifikáció is a term used in mathematics, particularly in topology, to describe the process of making a topological space compact. A compact space is a topological space where every open cover has a finite subcover. In simpler terms, a compact space is one that is "closed and bounded" in a way that generalizes to more abstract spaces.

The most common way to achieve compactness is by adding points to a space. When a space is made compact by adding points, it is called its compactification. There are several types of compactifications, depending on the properties of the original space and the desired properties of the resulting compact space.

A widely used type is the one-point compactification, also known as the Alexandroff compactification. This is achieved by adding a single point, often called the "point at infinity," to a locally compact Hausdorff space. For example, the real line R can be one-point compactified to a circle by adding a point at infinity. Similarly, n-dimensional Euclidean space Rn can be compactified to the n-dimensional sphere.

Another important type is the Stone-Čech compactification, which is the largest possible compactification of a Tychonoff space. It is unique for any given Tychonoff space and plays a significant role in functional analysis and measure theory. Other compactifications exist, such as the one-point compactification of non-locally compact spaces and various extensions related to uniform structures. The choice of compactification depends on the specific problem and the properties one wishes to preserve or achieve.