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This is an exercise from a topological book.

Show that the Sorgenfrey line and the Niemytzki plane are not homeomorphic.

Thanks for any help.

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Is the question really concerning Sorgenfrey line vs Niemytzki plane or should it maybe be Sorgenfrey plane vs. Niemytzki plane? – Sam Aug 9 '12 at 13:40
It is really concerning Sorgenfrey line vs Niemytzki plane. It is from Engelking's book. – Paul Aug 9 '12 at 13:50
up vote 2 down vote accepted

HINT: The Niemycki plane has a closed, discrete set of cardinality $2^\omega$. Does the Sorgenfrey line?

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Mm it don't seems to have such set, however its square seems to have such set. – Paul Aug 9 '12 at 13:32
That’s correct. To show that it does not have such a set, let $A$ be an uncountable subset of the Sorgenfrey line, and show that $A$ must have an accumulation point. – Brian M. Scott Aug 9 '12 at 13:42


Find a topological difference between the two. To do this you will have to think about the definitions of everything involved.

After you do that, be sure to add to your question to include what work you've done, so it doesn't sound so much like a demand.

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Thanks for your response! – Paul Aug 9 '12 at 13:33

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