I am having some difficulty with Rolfsen's derivation of the Wirtinger presentation of a knot in "Knots and Links" (pages 56 to 60). The basic setup is illustrated below.
The proof begins as follows.
Yet according to Rolfsen's exposition of Van Kampen's theorem (on pages 369 to 372), we require $A, B_1, …, B_n, C$ to be open. Clearly they are not.
I am new to algebraic topology, so I am probably missing something terribly obvious. Thanks!
Best Answer
You ask in the comments whether Van Kampen's Theorem works when all the sets in the given collection of subsets are deformation retracts of open sets. That's not quite enough, though.
Besides the hypothesis that each set in the given collection of subsets is open, every other hypothesis of Van Kampen's Theorem is a statement regarding properties of certain subcollections of the whole collection of subsets, specifically properties regarding: the path connectivity and fundamental group of the intersection of each such subcollection; and the inclusion induced homomorphism from the fundamental group of that intersection into the fundamental group of each subset of the subcollection.
So you will be fine if, in addition to requiring that each original set being a deformation retract of some corresponding open set, you also require that the intersection of each relevant subcollection of the original sets is a deformation retract of the intersection of the corresponding subcollection of open sets.
This extra requirement is true, for example, if your original collection consists of subcomplexes of some CW complex structure, as alluded in the comment of @KyleMiller.