“covering space”的概念、定义、习题解题方法及翻译-数学辞典

单词

covering space

释义

covering space

A covering space for a topological space X is a topological space C together with a continuous map (see continuous function) π:C → X, called the covering map, such that about each point p ∈ X is an open set U such that π−1(U) is a disjoint union of open sets in C which are homeomorphically mapped onto U by π. For example, the real line R is a covering space for the unit circle S1 with π:R → S1 given by π(x)=(cosx,sinx). The covering map winds R repeatedly onto S1. The pre-image of the point p = (1,0) is the set 2πℤ and so a small arc U around (1,0) exists which has pre-image made up of small disjoint intervals (2nπ−ε, 2nπ + ε).

A useful property of covering spaces is that if γ is a path in X there is a unique path Γ in C such that γ = π∘Γ. The path Γ is called the lift of γ.

A universal covering space is a covering space which is simply connected. So in the example above, R is a universal covering space for S1. A universal covering space of X is a covering space for any connected covering space of X. If a universal covering space exists, it is unique up to homeomorphism.

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