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The Dedekind property holds for the order of the points on any straight line.
It follows from axioms 1-12 by projection that the Dedekind property is true for all lines.
An open series is continuous if it is compact and possesses the Dedekind property.
Thus the definitions of compactness and of the Dedekind property can be at once transferred to a closed series.
These works of G. Cantor and Dedekind were of the greatest importance in the progress of the subject.