Sphere-sphere intersection is not a surface

In my topology lecture, my lecturer said that when two spheres intersect each other, the intersecting region is not a surface. Well, my own understanding is that the intersecting region should look like two contact lens combine together,back to back. The definition of a surface that I have now is that every point has a neighbourhood which is homeomorphic to a disc. I don’t see why the intersecting region is not a surface. It will be better if someone can guide me through picture.

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The intersection of two spheres is not “two contact lenses”: it is only the points in common, where two hollow spheres intersect. They intersect either at a single point (if they are tangent), or else their intersection is what we can envision as the boundary shared by two contact lenses that are joined back to back: i.e. a circle.

Think of it like this: If we were to glue two contact lens “back to back” $()$, we would only apply glue to their edges, because those are the only points at which they join or “connect”, and the points at which they join (i.e. intersect) is no more and no less than a circle (see, e.g., the image below). This is true, unless the lens are merely tangent: $\quad)(\quad$
in which case they intersect (“touch”) at that single point of tangency.

enter image description here

A sphere is the surface of a ball. The object you are thinking of is the surface of the intersection of two balls. But the intersection of the spheres is the intersection of the surfaces of those balls, which is not the same. Especially, each point in the intersection must be on both spheres. Now it is easy to see that most points on your lens-like object are only on one of the surfaces, but somewhere inside the other.

Indeed, the intersection of two spheres (assuming they intersect at all and are not identical) is always a circle (possibly a degenerate circle of radius 0, that is, a point, in case the spheres are tangential to each other). That circle is the edge of your lens-like object.

In general 2 spheres on $\mathbb{R}^3$ intersect on a circle which is a curve as you can simply imagine. You don’t need to see a picture to visualize that I think..