Question from last year's topology exam

[quasar987]

Question 1 says "Is the bouquet* of two 2-spheres a surface"?

How does this question even makes sense? A surface is a paracompact Hausdorff 2-manifold w/o boundaries, and a manifold is a topological space plus an atlas. Here, no atlas is provided!

* http://en.wikipedia.org/wiki/Bouquet_of_spheres#Examples

 

[matt grime]

You use the 'obvious' one. Let me put it this way - is S^2, the sphere a surface? Of cousre it is, but you would deny it was.

A surface is something that is locally 2-d. The bouquet of two circles is homeomorphic to the sphere S^2 with the equator squeezed to a point. I would say it was moot if that was s surface. It is obviously not a smoorth surface, and I susepct your question is implicitly assuming smoothness. I.e. smooth (orientable) surfaces are characterized up to homeomorphism by genus.

I think you're better off talking to the person who set the exam to ask them what they're getting at.