Definition: A topological space $X$ is called Hausdorff space if for each $x_1,x_2 \in X$ (they are distinct) we can always find neighborhoods $U_1,U_2$ of $x_1,x_2$ such that $U_1 \cap U_2 = \varnothing $.
If a space be Hausdorff with respect to a topology, is it Hausdorff with respect to others? And if no, why do we define a topological manifold to be Hausdorff without considering the induced topology by the differentiabe atlas ?