open sets in a regular surface in $R^3$

Question

I am reading Differential geometry of curves and surfaces by Do Carmo.

Let S be a regular surface in $R^3$.

I wonder

How is open set of S defined?

Is a subset of S open if and only if it is the intersection of some open set in $R^3 with S?

Answer 1

Here is the general definition of the subspace topology:

If $X$ in a subset of a topological set $Y$, an open set of $X$ for the subspace topology is the intersection of $X$ and an open set of $Y$.

Remark that this has nothing to do with differential topology.