Smooth Atlas of Differentiable Manifold M

[bigli]

can you be given a suitable smooth atlas to the subset M of plane that M to be a differentiable manifold? M={(x,y);y=absolute value of (x)}

 

[HallsofIvy]

Smells like homework (esp the "hint"!) so I'm moving this. Don't we always handle absolute value by looking at two separate cases? Give an atlas consisting of two open sets.

 

[bigli]

How am I use open sets for (0,0) of M?

 

[Reverie]

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.

 

[Reverie]

How am I use open sets for (0,0) of M?

Let T be the subspace topology of M inherited from the plane and let T' be the subspace topology of the x-axis inherited from the plane. Clearly, (x-axis, T') and (R,standard topology on the real line) are the same.

Projecting M onto the x-axis gives a bijective correspondence of M and the x-axis. This creates a 1-1 correspondence between the open sets comprising the topology T and the open sets comprising the topology T'.

 

[bigli]

problem is to find a suitable function that must to be differentiable itself and its inverse.but how function and its inverse to be differentiable at (0,0)?

 

[Reverie]

problem is to find a suitable function that must to be differentiable itself and its inverse.but how function and its inverse to be differentiable at (0,0)?

You have not reformulated the problem correctly. I suggest looking at the definition of a differentiable manifold and figuring out exactly what it is that you need to prove.