The aim of this paper is to show that every Hausdorff continuous interval-valued function on a completely regular topological space X corresponds to a Dedekind cut in C(X) and conversely.

Jachymski showed that the set $$\left\{(x,y)\in {𝐜}_0\times {𝐜}_0:{\left(\sum _{i=1}^n\alpha \left(i\right)x\left(i\right)y\left(i\right)\right)}_{n=1}^{\infty }\text{is}\text{bounded}\right\}$$ is either a meager subset of ${𝐜}_0\times {𝐜}_0$ or is equal to ${𝐜}_0\times {𝐜}_0$. In the paper we generalize this result by considering more general spaces than ${𝐜}_0$, namely ${𝐂}_0\left(X\right)$, the space of all continuous functions which vanish at infinity, and ${𝐂}_b\left(X\right)$, the space of all continuous bounded functions. Moreover, we replace the meagerness by $\sigma$-porosity.

Let $C_p\left(X\right)$ be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) $C_p\left(X\right)$ is Scott-domain representable; (b) $C_p\left(X\right)$ is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T₁-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that $C_p\left(X\right)$ is...