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Homering of continuous functions
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ring of continuous functions
Let $X$ be a topological space and $C(X)$ be the function space consisting of all continuous functions from $X$ into $\mathbb{R}$, the reals (with the usual metric topology).
Ring Structure on $C(X)$
To formally define $C(X)$ as a ring, we take a step backward, and look at $\mathbb{R}^{X}$, the set of all functions from $X$ to $\mathbb{R}$. We will define a ring structure on $\mathbb{R}^{X}$ so that $C(X)$ inherits that structure and forms a ring itself.
For any $f,g\in\mathbb{R}^{X}$ and any $r\in\mathbb{R}$, we define the following operations:
1. (addition) $(f+g)(x):=f(x)+g(x)$,
2. (multiplication) $(fg)(x):=f(x)g(x)$,
3. (identities) Define $r(x):=r$ for all $x\in X$. These are the constant functions. The special constant functions $1(x)$ and $0(x)$ are the multiplicative and additive identities in $\mathbb{R}^{X}$.
4. (additive inverse) $(f)(x):=(f(x))$,
5. (multiplicative inverse) if $f(x)\neq 0$ for all $x\in X$, then we may define the multiplicative inverse of $f$, written $f^{{1}}$ by
$f^{{1}}(x):=\frac{1}{f(x)}.$ This is not to be confused with the functional inverse of $f$.
All the ring axioms are easily verified. So $\mathbb{R}^{X}$ is a ring, and actually a commutative ring. It is immediate that any constant function other than the additive identity is invertible.
Since $C(X)$ is closed under all of the above operations, and that $0,1\in C(X)$, $C(X)$ is a subring of $\mathbb{R}^{X}$, and is called the ring of continuous functions over $X$.
Additional Structures on $C(X)$
$\mathbb{R}^{X}$ becomes an $\mathbb{R}$algebra if we define scalar multiplication by $(rf)(x):=r(f(x))$. As a result, $C(X)$ is a subalgebra of $\mathbb{R}^{X}$.
In addition to having a ring structure, $\mathbb{R}^{X}$ also has a natural order structure, with the partial order defined by $f\leq g$ iff $f(x)\leq g(x)$ for all $x\in X$. The positive cone is the set $\{f\mid 0\leq f\}$. The absolute value, given by $f(x):=f(x)$, is an operator mapping $\mathbb{R}^{X}$ onto its positive cone. With the absolute value operator defined, we can put a lattice structure on $\mathbb{R}^{X}$ as well:
Since taking the absolute value of a continuous function is again continuous, $C(X)$ is a sublattice of $\mathbb{R}^{X}$. As a result, we may consider $C(X)$ as a latticeordered ring of continuous functions.
Remarks. Any subring of $C(X)$ is called a ring of continuous functions over $X$. This subring may or may not be a sublattice of $C(X)$. Other than $C(X)$, the two commonly used latticeordered subrings of $C(X)$ are

$C^{*}(X)$, the subset of $C(X)$ consisting of all bounded continuous functions. It is easy to see that $C^{*}(X)$ is closed under all of the algebraic operations (ringtheoretic or latticetheoretic). So $C^{*}(X)$ is a latticeordered subring of $C(X)$. When $X$ is pseudocompact, and in particular, when $X$ is compact, $C^{*}(X)=C(X)$.
In this subring, there is a natural norm that can be defined:
$\f\:=\sup_{{x\in X}}f(x)=\inf\{r\in\mathbb{R}\midf\leq r\}.$ Routine verifications show that $\fg\\leq\f\\g\$, so that $C^{*}(X)$ becomes a normed ring.

The subset of $C^{*}(X)$ consisting of all constant functions. This is isomorphic to $\mathbb{R}$, and is often identified as such, so that $\mathbb{R}$ is considered as a latticeordered subring of $C(X)$.
References
 1 L. Gillman, M. Jerison: Rings of Continuous Functions, Van Nostrand, (1960).
Mathematics Subject Classification
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