euclidean topology

1. I Full-cone as topological space

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...
2. I About the definition of a Manifold

Hi, I'm a bit confused about the locally euclidean request involved in the definition of manifold (e.g. manifold ): every point in $X$ has an open neighbourhood homeomorphic to the Euclidean space $E^n$. As far as I know the definition of homeomorphism requires to specify a topology for...