Past exam of the topology course of the University of Oxford / 2014 / 1 / b / iii

Table of contents
- Solution iii

Solution

Any proper closed subset

F

of

Y

must be finite: if it were infinite, part (ii) would give

F = \overline{F} = Y

. Therefore every nonempty open subset of

Y

, being the complement of a proper closed set, is cofinite.

T_{Y} \subseteq T_{cofinite} .

The empty set belongs to both topologies. Inclusion need not be equality; the target may have fewer open sets.

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