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- Full-cone topological space but not topological manifold

Hello,

consider a full-cone (let me say a cone including bottom half, upper half and the vertex) embedded in ##E^3##. We can endow it with the topology induced by ##E^3## defining its open sets as the intersections between ##E^3## open sets (euclidean topology) and the full-cone thought itself as subset of ##E^3##. This way it has topological space structure.

Nevertheless I believe it is

Is that correct ? Thanks

consider a full-cone (let me say a cone including bottom half, upper half and the vertex) embedded in ##E^3##. We can endow it with the topology induced by ##E^3## defining its open sets as the intersections between ##E^3## open sets (euclidean topology) and the full-cone thought itself as subset of ##E^3##. This way it has topological space structure.

Nevertheless I believe it is

*not*a topological manifold because any neighborhood of the vertex cannot be homeomorphic to ##E^3## or one of its open subset.Is that correct ? Thanks