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We prove, as our main theorem, the finiteness of topological type of acomplete open Riemannian manifold $M$ with a base point $p \in M$ whose radialcurvature at $p$ is bounded from below by that of a non-compact model surfaceof revolution $\tilde{M}$ which admits a finite total curvature and has no pairof cut points in a sector. Here a sector is, by definition, a domain cut off bytwo meridians emanating from the base point $\tilde{p} \in \tilde{M}$. Noticethat our model $\tilde{M}$ does not always satisfy the diameter growthcondition introduced by Abresch and Gromoll. In order to prove the maintheorem, we need a new type of the Toponogov comparison theorem. As anapplication of the main theorem, we present a partial answer to Milnor's openconjecture on the fundamental group of complete open manifolds.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II