I'm reading Weeks' The Shape of Space. He mentions of intrinsic and extrinsic geometry and topology of surfaces (two-dimensional manifolds) in a vague but intuitive way.
Would it be correct to say that a change in intrinsic topology/geometry implies a change in extrinsic?
Also would a distinct extrinsic topology imply district extrinsic geometry?
By the extent to which I understand things,
(1) An intrinsic property is a property of the object itself, whereas an extrinsic property is a relation between the objects and other objects. This applies to other concepts, e.g. the number $2$ is even, eveness is an intrinsic property; the number $2$ is less than the number $3$, inferiority is an extrinsic property. Similarly, given a topological space $X$, $X$ must be either compact or non-compact, whereas we can only say $X$ is close/open assuming implicitly that $X$ is a subspace of another space $Y$.
(2) It depends on the nature the change. $2$ is even. If we plus it by $1$, it becomes odd, but it remains less than $5$.
(3) Geometry contains generally finer structure than topology. But I don't think there is any relation between the pair (intrinsic,extrinsic) and the pair (geometry, topology).