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 plane et al. were contractible, it is necessary to define a similar notion that holds for these non-metrizable surfaces.

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: For each c ∈ ℝ, take a disjoint copy H_c of the open upper half-plane, and one more copy called H. Now for each c ∈ ℝ, identify H with the subspace of H_c obtained by mapping each point p ∈ H to the point q ∈ H_c corresponding to pushing p directly away from c by exactly one unit. (I.e., q = p + (p-c)/∥p-c∥.) The image of H will be H_c minus a 2D semicircle of radius 1 about c. M is the resulting identification space. (M is clearly not maximal.)