Covering Space Definition

Algebraic Topology · Axiom Academy

LESSON Covering Space Definition The formal definition of covering maps and evenly covered neighborhoods 1. The Covering Map Definition A continuous surjection p: E → B is called a covering map (or covering projection ) if every point in the base space B has a neighborhood that is "evenly covered." 2. Evenly Covered Neighborhoods The key concept is that of an evenly covered neighborhood . If U is evenly covered by p, then p⁻¹(U) breaks into disjoint "sheets" that sit over U, each looking exactly like U. We write: p⁻¹(U) = ⊔ α∈A V α where each V α is open in E, the V α are pairwise disjoint, and p| V α : V α → U is a homeomorphism. The exponential map p: ℝ → S¹ defined by p(t) = e 2πit is the fundamental example of a covering space. The real line "wraps around" the circle infinitely many times. For any point on S¹, we can find a small arc U that doesn't wrap all the way around. Then p⁻¹(U) consists of infinitely many disjoint intervals in ℝ, each mapping homeomorphically to U. Covering maps have several important properties that make them useful in topology: Local homeomorphism: Each sheet V α is mapped homeomorphically to U Discrete fibers: For each b ∈ B, the fiber p⁻¹(b) is a discrete set in E Constant cardinality: If B is connected, all fibers have the same cardinality (the "number of sheets") Path lifting: Paths in B can be uniquely lifted to E (next lesson!)

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