If is an integer, , and and are vertices of a graph , then and are said to be -adjacent vertices of if there is a subgraph of , isomorphic to , containing and . A total -dominating set of is a set of vertices such that every vertex of is -adjacent to a vertex of . The total -domination number of is the minimum cardinality among the total -dominating sets of vertices of . It is shown that, for , if is a graph with no -isolated vertex, then . Further, -connectivity is defined and it is shown that, for , if is a -connected graph of order , then . We establish that the upper bounds obtained are best possible.