Compactification (mathematics)

In mathematics, compactification is a technique used to extend a space such that it becomes compact. Compactness is a topological property that has important implications in various areas of mathematics, particularly in analysis and topology. ### General Idea The process of compactification typically involves adding "points at infinity" or otherwise altering the topology of a space to ensure that every open cover of the space has a finite subcover.

The Alexandroff extension is a concept in topology, specifically in the study of topological spaces. It can be seen as a method to extend a given topological space by adding a "point at infinity," thereby creating a new space that retains certain properties of the original.

In topology, the concept of an "end" provides a way to classify the asymptotic behavior of a space at infinity. More formally, an end of a topological space can be understood as a way to describe how the space can be "accessed" from large distances.

In mathematics, particularly in the field of complex analysis, the term "prime end" refers to a concept used in the study of conformal mappings and, more generally, in the theory of Riemann surfaces and potential theory. The notion was introduced by the mathematician Henri Poincaré. **Prime Ends**: Prime ends can be thought of as a way to extend the notion of boundary points in a domain in the complex plane.

The Stone–Čech compactification is a mathematical concept in topology that extends a topological space to a compact space in a way that retains certain properties of the original space. It is named after mathematicians Marshall Stone and Eduard Čech. ### Definition Let \( X \) be a completely regular topological space.