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nrqed

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## Main Question or Discussion Point

This is a very simple topology question. Consider two infinite lines crossing at one point. Now, I know that this is not a 1D manifold, and I know the usual argument (in the neighbourhood of the intersection, we don't have a a line, or that if we remove the intersection point, we end up with four disconnected segments, which shows that we don't have a 1D manifold).

But what I want is to see how to prove that it is not a 1D manifold using only the basic definition. By the way, for now I want to focus on topological manifolds, not differentiable manifolds.

If I understand correctly, one has a 1D manifold if around all points we can have open sets that can be mapped to the real line.

Now let me consider an open set going from -y to y along one of the lines (so it contains the intersection point, at x=0,y=0) and an open set going from -x to x.

Let me define phi_1 to map the open set (-x,x) to the real line, with the intersection point mapped to the origin.

Similarly let me define phi_2 to map the open set (-y,y) to the real line with the intersection point mapped to the origin.

Now let me define the map phi that maps the neighbourhood of the intersection point to the real line in the following way:

Phi is defined to be phi_1 for the open set (-x,x) and it is defined to be phi_2 for the open set (-y,y).

My question is: this seems to map the neighbourhood of the intersection point to R1.

Now I know that two lines intersection is not, topologically, a 1D manifold. But what is the flaw in the above ??

Thanks in advance.

But what I want is to see how to prove that it is not a 1D manifold using only the basic definition. By the way, for now I want to focus on topological manifolds, not differentiable manifolds.

If I understand correctly, one has a 1D manifold if around all points we can have open sets that can be mapped to the real line.

Now let me consider an open set going from -y to y along one of the lines (so it contains the intersection point, at x=0,y=0) and an open set going from -x to x.

Let me define phi_1 to map the open set (-x,x) to the real line, with the intersection point mapped to the origin.

Similarly let me define phi_2 to map the open set (-y,y) to the real line with the intersection point mapped to the origin.

Now let me define the map phi that maps the neighbourhood of the intersection point to the real line in the following way:

Phi is defined to be phi_1 for the open set (-x,x) and it is defined to be phi_2 for the open set (-y,y).

My question is: this seems to map the neighbourhood of the intersection point to R1.

Now I know that two lines intersection is not, topologically, a 1D manifold. But what is the flaw in the above ??

Thanks in advance.