Let $X$ be a compact Hausdorff topological space such that for every continuous $f,g:X\to\mathbb{R}$ with $0\le f\le g$ there is a continuous $h:X\to\mathbb{R}$ such that $f=gh$.



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Is there a special term for such a property? Any characterizations in terms of the inner structure of $X$ or perhaps some properties of $C(X)$?


It is not hard to show that if $X$ is extremally disconnected this property is satisfied.


In my specific situation I have a compact space, but of course this property makes sense for an arbitrary topological space. Perhaps there is some information in this general case as well.



When can we divide continuous functions?
The kind of completely regular space you are looking for is an $F$-space.


Suppose that $X$ is a completely regular space. Then we say that a subset $A\subseteq X$ is $C^\*$-embedded if whenever $f:A\rightarrow\mathbb{R}$ is bounded and continuous, there is a continuous $g:X\rightarrow\mathbb{R}$ with $g|\_A=f$. We say that a subset of $X$ of the form $f^{-1}[\{0\}]^c$ for some continuous $f:X\rightarrow\mathbb{R}$ is a cozero set. An $F$-space is a space where every cozero set is $C^\*$-embedded.


Proposition: Let $X$ be a completely regular space. Then the following are equivalent:


1. $X$ is an $F$-space
2. whenever $f,g:X\rightarrow\mathbb{R}$ are continuous with $0\leq f\leq g$ there is some continuous $h:X\rightarrow\mathbb{R}$ with $f=gh$.
3. whenever $f,g:X\rightarrow\mathbb{R}$ are bounded and continuous with $0\leq f\leq g$ there is some continuous $h:X\rightarrow\mathbb{R}$ with $f=gh$.


Proof: $2\rightarrow 3$ follows since in (3) we strengthen the premises.


$1\rightarrow 2$ Suppose that $X$ is an $F$-space, and $0\leq f\leq g$. Then let $\ell:g^{-1}[\{0\}]^c\rightarrow\mathbb{R}$ be the function defined by letting
$\ell(x)=f(x)/g(x)$. Then $g^{-1}[\{0\}]^c$ is a cozero set, and $\ell$ is bounded, so $\ell$ extends to a bounded continuous function $h:X\rightarrow\mathbb{R}$. Observe that $f=gh$.


$3\rightarrow 1$. Let $U$ be a cozero set, and let $\ell:U\rightarrow[0,1]$ be a continuous function. Let $g:X\rightarrow[0,1]$ be a continuous function where $U=g^{-1}[\{0\}]^c$. Then let $f:X\rightarrow[0,1]$ be the continuous function such that $f(x)=g(x)\ell(x)$ whenever $x\in U$ and $f(x)=0$ elsewhere. We observe that $0\leq f\leq g\leq 1$, so there must be a continuous function $h:X\rightarrow[0,1]$ with $f=gh$. However, this means that $h(x)=\ell(x)$ whenever $x\in U$, so $h$ is a continuous extension of the function $\ell$.


Q.E.D.


There are plenty of other characterizations of the $F$-spaces. If $X$ is a completely regular space, then let $C(X)$ denote the ring of all continuous functions $f:X\rightarrow\mathbb{R}$. If $A,B\subseteq X$, then we say that $A,B$ are completely separated if there is a continuous function $f:X\rightarrow[0,1]$ with $A\subseteq f^{-1}[\{0\}],B\subseteq f^{-1}[\{1\}]$.


The following fact can be found in the standard text Rings of Continuous Functions by Gillman and Jerison.


Let X be a completely regular space. The following are equivalent:


1. $X$ is an $F$-space.
2. The Stone-Cech compactification $\beta X$ is an $F$-space.
3. Every finitely generated ideal in $C(X)$ is principal.
4. Every ideal in $C(X)$ is convex.
5. For all $f\in C(X)$, there is a $g\in C(X)$ with $f=g\cdot |f|$.
6. If $f\in C(X)$, then the sets $\{x\in X\mid f(x)>0\}$ and $\{x\in X\mid f(x)<0\}$ are completely separated.
7. If $M\subseteq C(X)$ is a maximal ideal, then the set of all prime ideals $P$ in $C(X)$ with $P\subseteq M$ is linearly ordered.
8. If $f,g\in C(X),$ then $\langle f,g\rangle=\langle|f|+|g|\rangle.$


**Comparison with other properties**


If $X$ is an $F$-space, then every countable subset of $X$ is $C^\*$-embedded. Therefore, every compact $F$-space has a homeomorphic copy of $\beta\omega$. A basically disconnected space is a completely regular space where every cozero set has open closure. Every basically disconnected space is an $F$-space, but $(\beta\omega)\setminus\omega$ is an $F$-space that is not basically disconnected. There are also some connected $F$-spaces.


