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Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome.

enter image description here

[Context of Image: related to the problem from this pdf document.]


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The picture by itself does not provide enough information. – Brian M. Scott Jan 8 '13 at 22:57
It's not. That space is homeomorphic to the Klein bottle. The lines and arrows are to show the identifications you make on $\mathbb R^2$ to get the Klein bottle. – Alex Becker Jan 8 '13 at 23:05
There’s more than one way to interpret the identifications. However, now that I’ve seen the PDF, I take it that each unit segment is to be identified with each other unit segment bearing the same kind of arrow. – Brian M. Scott Jan 8 '13 at 23:06
All of the interpretations that I had in mind involve identifying arrows in the same direction. The questions were whether the identification was by segment or by whole line, and whether the interiors of different squares were being identified as well as the edges. – Brian M. Scott Jan 9 '13 at 4:06
Yes; to get a Klien bottle, you have to collapse it down to one square and then identify the edges of that. – Brian M. Scott Jan 9 '13 at 14:35

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