A single cone attachment has a two-set open cover: 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. fully proved9Proved, and every prerequisite is fully proved too.
graph LR
n0("A single cone attachment has a two-set open cover"):::fully_proved
n1("An indexed family of cones has a two-set open cover"):::fully_proved
n2["The cone on a disk boundary is the disk"]:::fully_proved
n3["The base-side cone cover retracts onto the original space"]:::fully_proved
n4("The cone-side cover member is contractible"):::fully_proved
n5["The single-cone cover intersection has the homotopy type of its boundary"]:::fully_proved
n6("A single cone attachment is a topological pushout"):::fully_proved
n7["Collapsing the boundary of a one-disk gives the circle"]:::fully_proved
n8["The mapping cone of a topological pair"]:::fully_proved
n2 --> n7
n0 --> n2
n0 --> n1
n0 --> n3
n0 --> n5
n0 --> n6
n0 --> n4
n0 --> n7
n0 --> n8
n6 -.-> n8
class n0 focus
click n0 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-open-cover" "A single cone attachment has a two-set open cover — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#indexed-cone-open-cover" "An indexed family of cones has a two-set open cover — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#cone-disk-homeomorphism" "The cone on a disk boundary is the disk — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-base-retract" "The base-side cone cover retracts onto the original space — 1 item · 1 fully proved"
click n4 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-upper-contractible" "The cone-side cover member is contractible — 1 item · 1 fully proved"
click n5 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-intersection" "The single-cone cover intersection has the homotopy type of its boundary — 1 item · 1 fully proved"
click n6 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-pushout" "A single cone attachment is a topological pushout — 1 item · 1 fully proved"
click n7 "../../../../../roadmap/fundamental-group/van-kampen/presentation-complex/index.html#circle-one-cell-quotient" "Collapsing the boundary of a one-disk gives the circle — 1 item · 1 fully proved"
click n8 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-model" "The mapping cone of a topological pair — 1 item · 1 fully proved"