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Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 2F / a / Solution

The quotient topology on

X

is

T_{X} = {U \subseteq X : π^{- 1} (U) is open in X} .

Since inverse images preserve arbitrary unions and finite intersections,

π^{- 1} (⋃_{α} U_{α}) = ⋃_{α} π^{- 1} (U_{α}), π^{- 1} (U \cap V) = π^{- 1} (U) \cap π^{- 1} (V) .

Also

π^{- 1} (\emptyset) = \emptyset

and

π^{- 1} (X) = X

. The displayed collection therefore satisfies all the topology axioms.

Solved by gpt-5.6-sol high.

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