For the duration of this question, let a "surface" be any connected Hausdorff topological space that is locally homeomorphic to R2. Note that we make no assumption about a countable base to its topology.
For instance, if L denotes the long ray 𝜔1 × [0, 1] with the lexicographic order topology (𝜔1 is the first uncountable ordinal) and O denotes its endpoint, then we define the long plane P as L × S1 with {O} × S1 identified to a point.
Define a maximal surface as one that is not a proper subspace of any other surface.
Edit: Because I naïvely thought the long line et al. were contractible, it is necessary to define a similar notion that holds for these non-metrizable manifolds.
Definition: Call any surface jordan if every simple closed curve bounds a topological 2-disk D2 and the complement of this 2-disk is noncompact.
Then the long plane P is a jordan maximal surface.
Questions:
1. Are all jordan maximal surfaces homeomorphic to P?
2. Is it true that every jordan surface N is a subspace of a jordan maximal one?
(I suspect there is a proof with Zorn's Lemma, but the details are delicate and have so far escaped me.)
3. If 2. is true, is a maximal surface containing a given jordan surface N unique up to homeomorphism?
4. In particular: Consider the Prüfer manifold M.* If M is a subspace of a jordan maximal surface, what is its description, and is it unique up to homeomorphism?
* One way to define M: Take disjoint copies Hc of the open upper half-plane, one for each real number c, and one more copy called H. Now identify each Hc with the subspace of H obtained by mapping each point p ∈ Hc to the point q ∈ H corresponding to pushing p directly away from c by exactly one unit. (I.e., q = p + (p-c)/∥p-c∥.) The image of Hc will be H minus a 2D semicircle of radius 1 about c. M is the resulting identification space. (M is clearly not maximal.)