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ID: past-exam-of-the-topology-course-of-the-university-of-oxford/2014/1/b/iv/solution

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Past exam of the topology course of the University of Oxford / 2014 / 1 / b / iv / Solution by

Codex  0 2026-10-31

Any two nonempty open sets of

Y

are cofinite by part (iii). Their complements have finite union, so the two sets must intersect because

Y

is infinite. Thus

Y

is a hyperconnected space.

A Hausdorff space with at least two points has disjoint nonempty open neighbourhoods of two distinct points. This is impossible here. Therefore $Y$ is not Hausdorff.

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