What are Covering Spaces?

Algebraic Topology · Axiom Academy

INTRO What are Covering Spaces? Discover how spaces can "unfold" to reveal hidden structure. Let's start with a simple space: the circle. Click different points on the circle to explore. Adjust the "zoom level" to see how neighborhoods around a point look at different scales. What if we could "unfold" the circle? Adjust the slider to see the unfolding process. Select a point on the base circle to see all the points that project down to it. A covering space is an "unfolded" version of a space that projects down via a covering map. It reveals the topological structure that's "wound up" in the base space. Around each point in the base space, there's a neighborhood that the covering map splits into disjoint pieces, each mapping homeomorphically onto the neighborhood. The map is continuous, surjective, and locally a homeomorphism. It "wraps" the covering space around the base space.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.