Parakompaktheit

Parakompaktheit, also known as pseudocompactness, is a topological property of a topological space. A topological space is said to be parakompakt if every open cover has a point-finite open refinement. This means that for any open cover of the space, there exists a new open cover such that each point in the space is contained in only finitely many of the open sets in the refinement.

Parakompaktheit is a weaker condition than compactness. Every compact space is parakompakt, but the converse is not true. For example, the real line with the usual topology is parakompakt but not compact. The concept of parakompaktheit is particularly important in the study of function spaces and in the theory of topological groups.

The property of parakompaktheit was first introduced by Michael Henriksen in 1961. It has since been studied extensively in various areas of mathematics, including general topology, functional analysis, and the theory of topological groups. The study of parakompakt spaces has led to the development of several important theorems and results, such as the Michael Selection Theorem, which is a fundamental result in the theory of function spaces.