Compactness of first-order fuzzy logics

One of the nice properties of the first-order logic is the compactness of satisfiability. It states that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in fuzzy logics will pose various kinds of compactness in these logics.In this article, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in H\’{a}jek Basic logic, some new results are given around this issue.It will be shown that there are topologies on $[0,1]$ and $[0,1]^2$ for which the interpretation of all logical connectives of the Basic logic is continuous. Furthermore, a topology on first-order structures will be introduced for any similarity relation as well. Then by the same ideas as in continuous logic, the results around the compactness of satisfiability will be extended for Basic logic.