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.

Then the long plane P is a maximal contractible surface.

Edit: This may require redefining "contractibility" to mean something like "The complement of some point p is topologically foliated by 1-manifolds, all emanating from p." Suddenly I am not even sure if the long ray is contractible by the usual definition, no less the long plane.

Questions:

1. Are all maximal contractible surfaces homeomorphic to P?

2. Is it true that every contractible surface N is a subspace of a 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 for any given contractible surface N unique up to homeomorphism?

4. In particular: Consider the Prüfer manifold M.^* If M is a subspace of a maximal contractible surface, what is its description, and is it unique up to homeomorphism?

* One way to define M: Take disjoint copies H_c of the open upper half-plane, one for each real number c, and one more copy called H. Now identify each H_c with the subspace of H obtained by mapping each point p ∈ H_c 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 H_c will be H minus a 2D semicircle of radius 1 about c. M is the resulting identification space. (M is clearly not maximal.)