Since Halls of Ivy designated this as a homework problem, I'm not sure what the appropriate protocol is. I am new to these forums.
Let me rephrase the problem so that it is slightly clearer.
Let M={(x,y)|y=|x|}. M inherits the subspace topology from the plane. Denote the subspace topology by T. Does there exist a smooth altas A on M making M into a 1-dimensional smooth differentiable manifold such that the topology induced by the smooth atlas is T?
The answer is yes. To aid your understanding, I recommend doing the following.
1.)Convince yourself that every smooth atlas on a set M gives rise to a topology on M. Here, I am assuming that a manifold structure is introduced on a set and not a topological space. Some authors start by introducing a manifold structure on a topological space that is Hausdorff and second countable. If one does this, then the topology induced by the smooth atlas coincides with the original topology on M. If one introduces the manifold structure on an arbitrary set, the induced topology is not necessarily Haussdorff or second countable.
2.)Convince yourself that there exists a smooth atlas on every set that has the same cardinality as the real line. This proves that M has a manifold structure. However, the induced topological structure doesn't necessarily coincide with the subspace topology T on M. The goal of this problem should be to find a manifold structure which induces the subspace topology T on M.
3.)Using the projection of M onto the x-axis, define a smooth atlas on M. Defining a smooth atlas in this way produces a smooth structure that induces the subspace topology T on M. You need to prove the statements that I asserted in this step.
