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A sphere has no edges or edges but its face is globular.
a cylinder
A cone has Zero verticies. no a cone has one verticie the point at the top counts as a verticie
An octagon, (2D) has one face, zero edges, and eight sides.
Your question is inconsistent on its face. You asked "What shape has zero faces . . . . . and has only one face." I'd say that nothing could ever meet both of those requirements. They are ... how you say ... 'mutually exclusive'.