coordinate systems produce submanifolds

Question

In his book "Semiriemannian geometry with applications to relativity", Barrett O'Neil says on page 16 under definition 26 that "coordinate systems produce submanifolds.If T:U->R^n is a coordinate system in a manifold M then holding any n-m of the coordinate functions constant produces an m- dimensional submanifold." My question is ,is this submanifold the same as the inverse image of [R^m×{0}] under T? If yes then since this set is open in M it should be a manifold of the same dimension. This is open in M because T is a homeomorphism and [R^m×{0}] is open in R^n. Where am i going wrong and what is an explicit description of this submanifold? Thanks.