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each compact set has a compact pre-image

Source: Iran PPCE 2012-Analysis exam-P4

February 14, 2012

topologyreal analysisreal analysis unsolved

Problem Statement

Suppose that

X

and

Y

are metric spaces and

f:X \longrightarrow Y

is a continious function. Also

f_1: X\times \mathbb R \longrightarrow Y\times \mathbb R

with equation

f_1(x,t)=(f(x),t)

for all

x\in X

and

t\in \mathbb R

is a closed function. Prove that for every compact set

K\subseteq Y

, its pre-image

f^{pre}(K)

is a compact set in

X