Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 21F (Linear Analysis)

The Tietze extension theorem states: if $X$ is a normal topological space, $A\subseteq X$ is closed, and $f:A\to [-M,M]$ is continuous, then there is a continuous $F:X\to [-M,M]$ with $F{|}_A=f$.

We first prove an approximation lemma. Given continuous $h:A\to [-r,r]$, the closed subsetsare disjoint and closed in $X$. By the Urysohn lemma, there is a continuous $g:X\to [-r/3,r/3]$ equal to $-r/3$ on $A_{-}$ and $r/3$ on $A_{+}$. Then

Starting with $h_0=f$ and $r_0=M$, apply the lemma recursively to obtain continuous $g_j:X\to \mathbb{R}$ such thatwhere $r_j=M(2/3{)}^j$. The seriesconverges uniformly by the Weierstrass test, so its sum is continuous. On $A$, its remainder after $j$ terms is $h_{j+1}$, which tends uniformly to zero; hence $F{|}_A=f$. The bounds also keep $F$ in $[-M,M]$ after the standard endpoint-preserving version of the construction, proving the theorem.

Writewith each $U_n$ open. The closed sets $A$ and $X\setminus U_n$ are disjoint. By the Urysohn lemma, choose a continuous $f_n:X\to [0,1]$ withThe uniformly convergent seriesdefines a continuous map to $[0,1]$. It vanishes on $A$. If $x\notin A$, then $x\notin U_n$ for some $n$, so $f_n(x)=1$ and $f(x)>0$. Thuswhich is the closed G-delta set as a zero set construction.

Apply part b separately to obtain continuous $g,h:X\to [0,1]$ withBecause $A\cap B=\varnothing$, the two functions never vanish simultaneously. Henceis continuous and takes values in $[0,1]$. It equals zero exactly where $g=0$, namely on $A$, and equals one exactly where $h=0$, namely on $B$.