Hello,
The open long ray . My question is how can we show that the open long ray is Hausdorff locally homeomorphic to the real line , so that it is a manifold of 1-dimension. Also, how can we show that it is normal but not metrizable.
Please guide me
Thank you in advance
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 is an open cover of having no countable subcover.
But I miss two points, I should also verify that the open long ray is connected and locally homeomorphic to , in order to say that it is a manifold. How can I show that.
Thank you very much
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.
Thank you in advance