Universal covering

The universal covering $ˆ 𝑝 : ˆ 𝑋 \to 𝑋$ is a covering with a simply connected covering space $ˆ 𝑋$ . #m/def/homotopy It follows immediately that the characteristic subgroup of $ˆ 𝑝$ is trivial, and by deck transformation group of a regular covering as quotient

A u t 𝖢 𝗈 𝗏 𝑋 (𝑝) ≅ 𝜋 1 (𝑋, 𝑥 0)

for any $𝑥 0 \in 𝑋$ . The universal covering is universal in the sense that $ˆ 𝑝 : (ˆ 𝑋, ˆ 𝑥 0) ↠ (𝑋, 𝑥 0)$ is the initial object of the category of pointed connected coverings $𝖢 𝗈 𝗏 (𝑋, 𝑥 0)$ , assuming $𝑋$ is locally path-connected.

Properties

The universal covering is regular.
Main theorem of coverings
A locally path-connected, connected space is semilocally simply connected iff it has a universal covering.

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