Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 21F / b

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- Solution b

Solution

Write

A = ⋂_{n = 1}^{\infty} U_{n}

with each

U_{n}

open. The closed sets

A

and

X ∖ U_{n}

are disjoint. By the Urysohn lemma, choose a continuous

f_{n} : X \to [0, 1]

with

f_{n} = 0 on A, f_{n} = 1 on X ∖ U_{n} .

The uniformly convergent series

f (x) = \sum_{n = 1}^{\infty} 2^{- n} f_{n} (x)

defines a continuous map to

[0, 1]

. It vanishes on

A

. If

x \in / A

, then

x \in / U_{n}

for some

n

, so

f_{n} (x) = 1

and

f (x) > 0

. Thus

f^{- 1} (0) = A,

which is the closed G-delta set as a zero set construction.

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