Topological space maps naturally to max-spectrum of ring of continuous real-valued functions

Statement

Suppose $X$ is a topological space. Let $C(X,\mathbb {R} )$ denote the ring of continuous real-valued functions on $X$. Then, there is a natural injective (set-theoretic) map to the max-spectrum $\operatorname {Max-Spec} (C(X,\mathbb {R} ))$:

$X\to \operatorname {Max-Spec} (C(X,\mathbb {R} ))$

given as follows:

$x\mapsto M_{x}=\{f\in C(X,\mathbb {R} )\mid f(x)=0\}$.

Proof

Given: A topological space $X$, the ring $C(X,\mathbb {R} )$ of continuous real-valued functions on $X$. A point $x\in X$. $M_{x}=\{f\in C(X,\mathbb {R} )\mid f(x)=0\}$.

To prove: $M_{x}$ is a maximal ideal in $C(X,\mathbb {R} )$.

Proof: Consider the map:

$\operatorname {eval} _{x}:C(X,\mathbb {R} )\to R$

given by:

$\operatorname {eval} _{x}(f)=f(x)$.

By the definition of pointwise addition and multiplication, $\operatorname {eval} _{x}$ is a homomorphism from $C(X,\mathbb {R} )$ to $\mathbb {R}$. Further, since constant functions are continuous, every $c\in \mathbb {R}$ can be expressed as $\operatorname {eval} _{x}(f)$ where $f$ is the constant function $c$. Thus, $\operatorname {eval} _{x}$ is surjective. Thus, $\operatorname {eval} _{x}$ is a surjective homomorphism to a field, so its kernel, which is $M_{x}$, is a maximal ideal.