# The open long ray

• Jun 4th 2011, 02:04 AM
student2011
The open long ray
Hello,

The open long ray $\displaystyle L^{+}=w_{1}\times[0,1)-{(0,0)}$. My question is how can we show that the open long ray is Hausdorff locally homeomorphic to the real line $\displaystyle R$, so that it is a manifold of 1-dimension. Also, how can we show that it is normal but not metrizable.

• Jun 4th 2011, 04:54 PM
Drexel28
Quote:

Originally Posted by student2011
Hello,

The open long ray $\displaystyle L^{+}=w_{1}\times[0,1)-{(0,0)}$. My question is how can we show that the open long ray is Hausdorff locally homeomorphic to the real line $\displaystyle R$, so that it is a manifold of 1-dimension. Also, how can we show that it is normal but not metrizable.

What have you tried so far--this is just a get down and dirty problem if you know what I mean?
• Jun 7th 2011, 03:16 AM
student2011
What I tried is the following:

1. The open long ray is normal since all ordered spaces are completely normal.

2. The open long ray is not metrizable, since it is not Lindelof ( the collection $\displaystyle {(0,\alpha) \alpha \in \omega _{1}}$ is an open cover of $\displaystyle L^{+}$ having no countable subcover.

But I miss two points, I should also verify that the open long ray is connected and locally homeomorphic to $\displaystyle R$, in order to say that it is a manifold. How can I show that.

Thank you very much
• Jun 11th 2011, 02:29 PM
student2011
The open long ray
Hello;

I am now studying the open long ray, and I proved some properties, for example the open long ray is path connected, hence connected. Kindly, I want from you to check if what I wrote is logic or not.

Please find attached to my work.