For the duration of this question, let a "surface" be any connected Hausdorff topological space that is locally homeomorphic to **R**<sup>2</sup>. *Note that we make no assumption about a countable base to its topology.*

For instance, if **L** denotes the long ray 𝜔<sub>1</sub> × [0, 1] with the lexicographic order topology (𝜔<sub>1</sub> is the first uncountable ordinal) and *O* denotes its endpoint, then we define the *long plane* **P** as **L** × **S**<sup>1</sup> with {*O*} × **S**<sup>1</sup> 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**.<sup>*</sup> *If* **M** *is a subspace of a maximal contractible surface, what is its description, and is it unique up to homeomorphism?*
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\* One way to define **M**: Take disjoint copies **H**<sub>c</sub> of the open upper half-plane, one for each real number c, and one more copy called **H**. Now identify each **H**<sub>c</sub> with the subspace of **H** obtained by mapping each point p ∈ **H**<sub>c</sub> 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**<sub>c</sub> will be **H** minus a 2D semicircle of radius 1 about c. **M** is the resulting identification space. (**M** is clearly not maximal.)