Determine whether or not the topological space Y x Y (the cartesian product of two Y-shaped spaces, each the union of 3 closed intervals) can be embedded as a subset of 3-space.

it cannot be embedded. Y x Y is homeomorphic to the cone over the graph C_3 x C_3 . if there were such an embedding, intersecting with a small sphere centered at the vertex (the point (y_0, y_0) in Y x Y , where y_0 is the branch point of Y ) would give an embedding of C_3 x C_3 in the sphere. this is impossible, because this graph is non-planar. mike