The authors obtain bounds for the small and large inductive dimension of a space in terms of the dimensions of two subspaces, without the requirement of hereditary normality. In Section 2, a number of Urysohn inequalities are obtained and these are applied in Section 3 to obtain addition theorems. Typical of the results of Section 3 is the following: If \(X=X_1\cup X_2\) where Ind\(\,X_1=m\) and Ind\(\,X_2=n\), then ind\(\, X\leq f(m,n)\), where \(f(m,-1)=m\), \(f(-1,n)=n\) and for \(m,n\geq 0\), \(f(m,n)=\text{ max}\{f(m-1,n), f(m,n-1)\}+\text{ max}\{m,n\}+3\). More precise bounds are given in case \(X\) is normal. In Section 4, the authors study the dimension functions Ind\(_0\) and ind\(_0\) first defined by A. V. Ivanov [Mosc. Univ. Math. Bull. 31, 64–69 (1976; Zbl 0351.54028)] (but attributed to Filippov) and independently by the first author of this paper [Ann. Univ. Sci. Budap. Eötvös, Sect. Math. 18(1975), 15–25 (1976; Zbl 0332.54028)]. Among other results in this section they construct a completely normal compact space \(X\) which is the disjoint union of two dense 0-dimensional (in all the usual senses) subsets but for which ind\(_0\,X=\text{ Ind}_0\, X=\infty\).