A space \(X\) is submaximal if every dense subspace of \(X\) is open in \(X\). If \(X\) is a \(T_0\)-space and \(x,y\in X\) then \(x\leq y\) if and only if \(y\in \overline{\{x\}}\); this order is called a specialization order on \(X\). Call a set \(\{x_0,\dots, x_n\}\subset X\) a chain if \(x_0<\dots< x_n\); in this case \(n\) is called the length of the chain. The supremum of the lengths of chains in \(X\) is the Krull dimension of \(X\). The paper contains several examples and statements showing the relationship of submaximality with the Krull dimension being less than or equal to one. It is proved, among other things, that a compact Hausdorff space is submaximal if and only if it has finitely many non-isolated points. It is also established that the one-point compactification of a space \(X\) is submaximal if and only if every dense set \(U\subset X\) is open and \(X\setminus U\) is compact.