In the book Riemannian Geometry, Do Carmo gave his definition of manifold without metionning to the topology on the set M. However, the usual way to define manifold is always to supposing M is a topological space with some restrictions on the topology, such as $A_{2}$ or $T_{2}$.
Later, he made a remark on this. As he said, we can induce a natural topology on M using the differential structure.
My question is that:
What happened if the topology induced in this way may not be $T_{2}$ or $A_{2}$? Do Carmo metioned this problem but he didn't give an answer. I am in a mess now...Help, thanks in advance.
