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.

>*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 D<sup>2</sup> 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**.<sup>*</sup> *If* **M** *is a subspace of a jordan maximal surface, what is its description, and is it unique up to homeomorphism?*
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\* One way to define **M**: For each c ∈ ℝ, take a disjoint copy **H**<sub>c</sub> of the open upper half-plane, and one more copy called **H**. Now for each c ∈ ℝ, identify **H** with the subspace of **H**<sub>c</sub> obtained by mapping each point p ∈ **H** to the point q ∈ **H**<sub>c</sub> 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**<sub>c</sub> minus a 2D semicircle of radius 1 about c. **M** is the resulting identification space. (**M** is clearly not maximal.)