Topological Spaces-Hausdorff Spaces

المؤلف:  David R. Wilkins

المصدر:  Algebraic Topology

الجزء والصفحة:  4-5

24-9-2016

1702

Example :The Zariski topology on the set R of real numbers is defined as follows: a subset U of R is open (with respect to the Zariski topology) if and only if either U = ∅ or else    R U is finite. It is a straightforward exercise to verify that the topological space axioms are satisfied, so that the set R of real numbers is a topological space with respect to this Zariski topology. Now the intersection of any two non-empty open sets in this topology is always non-empty. (Indeed if U and V are non-empty open sets then U = R F₁and V = R F2, where F₁ and F₂ are finite sets of real numbers. But then U ∩ V = R (F₁ ∪ F₂), which is non-empty, since F₁ ∪ F₂ is finite and R is infinite.) It follows immediately from this that R, with the Zariski topology,  is not a Hausdorff space.