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  "body": "\n## Summary\n\nA covering space of a space $X$ is a space $E$ with a continuous map $p \\colon E \\to X$ of a special kind.\nOver a small neighbourhood of each point of $X$, the map looks like a stack of disjoint copies of that neighbourhood.\nSuch a map $p$ is a covering map.\nCovering spaces have the path lifting and homotopy lifting properties, and for nice spaces they are classified by subgroups of the [[fundamental group|S1XK3HXHFX]].\nThey give the computation of $\\pi_1(S^1)$ and link topology with group theory.\n\n## Explanations\n\n### L1 human\n\nPicture an infinite spiral staircase above a circle, with each step projecting straight down onto the floor.\nNear each point of the circle, the part of the staircase above it is a pile of separate small pieces, one on each turn.\nThat is the covering $\\R \\to S^1$, $t \\mapsto e^{2\\pi i t}$.\nAnother example is the map $z \\mapsto z^2$ from the circle to itself, which wraps the circle twice around itself.\n",
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