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Two questions on Sorgenfrey line

This is an exercise from Engelking's Book.

Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$, but cannot be mapped onto $D(2^\omega)$?

I don't how to show it. Could anybody help me? Thanks ahead.

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    And also this: [On deleting vs. closing duplicate questions.](http://meta.math.stackexchange.com/questions/3338/on-deleting-vs-closing-duplicate-questions). All three can be briefly summarized in a way, that it is frowned upon.2012-08-09

1 Answers 1

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HINTS:

  1. The Sorgenfrey line is the union of the sets $[n,n+1)$ for $n\in\Bbb Z$.

  2. The Sorgenfrey line is separable. A continuous map onto $D(2^\omega)$ would contradict this; how?

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    @Brain Thanks. I will go.2012-08-10