The subgroup projection is a path-connected covering: local context

Project map · Chapter map · Full theorem DAG · Open textbook statement

This local map shows one dependency hop in either direction. The highlighted item is the current focus. What the colours mean

fully proved6Proved, and every prerequisite is fully proved too.

graph LR n0["The covering space associated to a subgroup"]:::fully_proved n1("The sets U[γ] form the universal-cover basis"):::fully_proved n2("The endpoint map is a covering"):::fully_proved n3["The path-class cover is path-connected"]:::fully_proved n4("The subgroup projection is a path-connected covering"):::fully_proved n5("The subgroup cover realizes the chosen subgroup"):::fully_proved n4 -.-> n5 n0 --> n5 n0 --> n4 n1 -.-> n4 n1 -.-> n2 n1 -.-> n3 n2 -.-> n4 n3 -.-> n4 click n0 "../../../../roadmap/fundamental-group/covering-spaces/index.html#subgroup-cover-space" "The covering space associated to a subgroup — 1 item · 1 fully proved" click n1 "../../../../roadmap/fundamental-group/covering-spaces/universal-cover/index.html#universal-cover-basis" "The sets U[γ] form the universal-cover basis — 1 item · 1 fully proved" click n2 "../../../../roadmap/fundamental-group/covering-spaces/universal-cover/index.html#universal-cover-is-covering" "The endpoint map is a covering — 1 item · 1 fully proved" click n3 "../../../../roadmap/fundamental-group/covering-spaces/universal-cover/index.html#universal-cover-path-connected" "The path-class cover is path-connected — 1 item · 1 fully proved" class n4 focus click n4 "../../../../roadmap/fundamental-group/covering-spaces/index.html#subgroup-cover-is-covering" "The subgroup projection is a path-connected covering — 1 item · 1 fully proved" click n5 "../../../../roadmap/fundamental-group/covering-spaces/index.html#subgroup-cover-image" "The subgroup cover realizes the chosen subgroup — 1 item · 1 fully proved"