Are the Rational numbers an open subset of the Real numbers with respect to the Sorg. topology?
No really sure how to begin. Can somebody start me off?
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An open set is a union of basis elements. Is there any basis element of that topology that doesn't contain any irrational numbers?
Well, am I correct in saying that this is the basis:
Surely you have the definitions available in your textbook or lecture notes?
No, lecturer is the worst. I'm having a hard enough time finding info off of google without it being way too complicated.
okay here is a basis, correct?
So, in answer to your question, all of the intervals contain irrational numbers?
Now you seem to be putting a topology on R^2 (Sorgenfrey plane). What is a rational number in R^2? Do you mean elements of QxQ? If so, then the original hint applies and yes, your answer to the question is correct. Can you finish from there?
To be absolutely honest, I don't know what to do next.
If every set in a collection contains non-rationals, then any union of those sets must contain non-rationals. Every open set is such a union, and so...?
and so the set of rationals cannot be an open subset?
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