 Solution

ID: past-exam-of-the-analytic-topology-and-axiomatic-set-theory-course-of-the-university-of-oxford/2010/b/5/d/i/solution

Constant functions give the lower bound

ℵ_{2} \leq ℵ_{2}^{ℵ_{0}}

ℵ_{2} = 2^{ℵ_{1}}

ℵ_{2}^{ℵ_{0}} = (2^{ℵ_{1}})^{ℵ_{0}} = 2^{ℵ_{1} \cdot ℵ_{0}} = 2^{ℵ_{1}} = ℵ_{2} .

Hence

ℵ_{2}^{ℵ_{0}} = ℵ_{2}

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