Generalized Lefschetz numbers of pushout maps defined on non-connected spaces

Let A, ${\displaystyle X_1}$ and ${\displaystyle X_2}$ be topological spaces and let ${\displaystyle i_1:A\to X_1}$, ${\displaystyle i_2:A\to X_2}$ be continuous maps. For all self-maps ${\displaystyle f_A:A\to A}$, ${\displaystyle f_1:X_1\to X_1}$ and ${\displaystyle f_2:X_2\to X_2}$ such that ${\displaystyle f_1{i}_1=i_1{f}_A}$ and ${\displaystyle f_2{i}_2=i_2{f}_A}$ there exists a pushout map f defined on the pushout space ${\displaystyle X_1{\bigsqcup }_A{X}_2}$. In [F] we proved a formula relating the generalized Lefschetz numbers of f, ${\displaystyle f_A}$, ${\displaystyle f_1}$ and ${\displaystyle f_2}$. We had to assume all the spaces involved were connected because in the original definition of the generalized Lefschetz number given by Husseini in [H] the space was assumed to be connected. So, to extend the result of [F] to the not necessarily connected case, a definition of generalized Lefschetz number for a map defined on a not necessarily connected space is given; it reduces to the original one when the space is connected and it is still a trace-like quantity. It allows us to prove the pushout formula in this more general case and therefore to get a tool for computing Nielsen and generalized Lefschetz numbers in a wide class of spaces.