Applications of (Proximal) Taimanov Theorem

Let $P^*(X)$ be the algebra of bounded, real-valued proximally continuous functions on an $EF$-proximity space $(X, \delta)$, where $X$ is a dense subspace of a Tychonoff topological space $S$. Mattson obtained several conditions which are equivalent to the following property: every member of $P^*(X)$ has a continuous extension to $S$. In this paper, we generalize the above problem to $L$-proximity via proximal Taimanov theorem when $S$ is a $T_1$ space.