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

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

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