We know the celebrated 'Whitney embedding theorem' for smooth manifold that says any n-dimensional manifold can be smoothly embedded in $\ \mathbb R^{2n} \ $. Now my question is: Is there similar version for topological manifolds? that is for which least N(n) any n-dim topological manifold will have a locally flat embedding in $\ \mathbb R^{N(n)} \ $. It's clear that for smoothable topological manifolds N will be equal to 2n for $\ \mathbb RP^{2n} \ $ can not be embedded(locally flat) in $\ \mathbb R^{2n-1} \ $ ie. $\ \mathbb RP^{2n} \ $ can not be realized as topological submanifold of $\ \mathbb R^{2n-1} \ $ where n is a power of 2.
But for nonsmoothable topological manifolds can this N be made less?