Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

Solution 1:

The following answers a weaker form of your question, namely, is it true that every connected topological manifold contains an open dense simply-connected subset? The answer to this question is positive and even more is true:

Theorem. Every connected topological $n$-manifold contains an open dense subset homeomorphic to $R^n$.

This follows from a theorem by R.Berlanga "A mapping theorem for topological sigma-compact manifolds", Compositio Math, 1987, vol. 63, 209-216.

Berlanga's theorem generalized an earlier work by M.Brown, who proved the same theorem for compact topological manifolds. (The case of triangulated manifolds is easy but serves as a guideline for proofs in the topological category.)

Berlanga's theorem does not answer, however, the question if every maximal simply-connected open subset is dense in $M$.