A Remark on Regular Points of Ricci Limit Spaces

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Let Y be a Gromov-Hausdorff limit of complete Riemannian n-manifolds with Ricci curvature bounded from below. A point in Y is called k-regular, if its tangent is unique and is isometric to an k-dimensional Euclidean space. By B5, there is k>0 such that the set of all k-regular point Rₖ has a full renormalized measure. An open problem is if Rₗ= for all l<k? The main result in this paper asserts that if R₁≠, then Y is a one dimensional topological manifold. Our result improves the Handa's result Honda that under the assumption that 1≤ dimₕ(Y)<2.