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In this article we study the comaximal graph ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$ of the ring $C\left(X\right)$. We have tried to associate the graph properties of ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$, the ring properties of $C\left(X\right)$ and the topological properties of $X$. Radius, girth, dominating number and clique number of the ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$ are investigated. We have shown that $2\le Rad{\Gamma }_{{}_2}^{\text{'}}C\left(X\right)\le 3$ and if $\left|X\right|>2$ then $\mathrm{girth}{\Gamma }_{{}_2}^{\text{'}}C\left(X\right)=3$. We give some topological properties of $X$ equivalent to graph properties of ${\Gamma }_{{}_2}^{\text{'}}C\left(X\right)$. Finally we have proved that $X$ is an almost $P$-space which does not have isolated points if and only if $C\left(X\right)$ is an almost regular ring...

Let $I$ be a semi-prime ideal. Then $P_{\circ }\in Min\left(I\right)$ is called irredundant with respect to $I$ if $I\ne {\bigcap }_{P_{\circ }\ne P\in Min\left(I\right)}P$. If $I$ is the intersection of all irredundant ideals with respect to $I$, it is called a fixed-place ideal. If there are no irredundant ideals with respect to $I$, it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point $p\in \beta X$ is a fixed-place point if $O^p\left(X\right)$ is a fixed-place ideal. In this situation...