Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / E / 2 / ii

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- Solution ii

Solution

If the inclusion is continuous from

(A, T^{'})

, then

i^{- 1} (U) = A \cap U

belongs to

T^{'}

for every

U \in T

. These intersections are precisely all members of

T_{A}

, so

T_{A} \subseteq T^{'} .

Thus the subspace topology is the coarsest topology on

A

making the inclusion continuous.

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