Geodesic
A Geometric Approach to the Heine-Borel Theorem
Canadian journal of mathematics, Tome 23 (1971) no. 5, pp. 845-848

Voir la notice de l'article provenant de la source Cambridge University Press

In a topological plane with strong enough topological properties one can use [6] open triangular regions to define a base for the topology. Similarly, one can use these regions to define boundedness of a set. In this setting we show that in the absolute plane geometry, the holding of the Heine-Borel theorem is equivalent to every four points being contained in some such region and that this second condition is equivalent to the parallel postulate. Thus we give two new conditions equivalent to the parallel postulate. 
Killgrove, R. B.; Frand, Jason; Giles, William; Bray, Henry. A Geometric Approach to the Heine-Borel Theorem. Canadian journal of mathematics, Tome 23 (1971) no. 5, pp. 845-848. doi: 10.4153/CJM-1971-094-1
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