Algebra

Stone–Čech compactification One of the proof of the Stone–Čech compactification is to consider the continuous functions from a given topological space $X$ to $[0, 1]$. Crux 1: constructing $[0, 1]^{C}$ Let $C$ be the space of all continuous functions from $X$ to $[0, 1]$, consider $[0, 1]^{C}$, there is a natural map from $X$ to $[0, 1]^{C}$: for each $x \in X$, define $\phi(x) = f \mapsto f(x)$. With product topology $\phi$ is continuous. Also by Tychonnof’s theorem $[0, 1]^{C}$ is compact. Now the closure of the image is obviously compact Hausdorff, and it satisfies a nice property: any continuous map $f : X \to [0, 1]$ extends uniquely to a continuous map $\overline{\phi(X)} \to [0,1]$: for any $\phi(x)$, it has to map to $f(x)$! ...

I am reviewing some fundamental algebra, and I just learnt something beautifully suggesting the connection of ring theory and geometry, which made me eager to learn some algebraic geometry (which I was not very interested when I was in undergrad). It is an exercise in Aluffi’s book1. Lets’ jump into it. The problem Let $K$ be a compact topological space, and let $R$ be the ring of continuous real-valued functions on $K$, with addition and multiplication defined pointwise. (i) For $p \in K$, let $M_p = \{f \in R | f(p) = 0\}$. Prove that $M_p$ is a maximal ideal in $R$. (ii) Prove that if $f_1, \dots , f_r \in R$ have no common zeros, then $(f1, \dots , fr)$ = $(1)$. (Hint: Consider $f^2_1 + \dots + f^2_r$ .) (iii) Prove that every maximal ideal $M$ in $R$ is of the form $M_p$ for some $p \in K$. (Hint: You will use the compactness of $K$ and (ii).) ...

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