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Vector field guided path following (VF-PF) algorithms are fundamental in robot navigation tasks, but may not deliver the desirable performance when robots encounter singular points where the vector field becomes zero. The existence of singular points prevents the global convergence of the vector field's integral curves to the desired path. Moreover, VF-PF algorithms, as well as most of the existing path following algorithms, fail to enable following a self-intersected desired path. In this paper, we show that such failures are fundamentally related to the mathematical topology of the path, and that by "stretching" the desired path along a virtual dimension, one can remove the topological obstruction. Consequently, this paper proposes a new guiding vector field defined in a higher-dimensional space, in which self-intersected desired paths become free of self-intersections; more importantly, the new guiding vector field does not have any singular points, enabling the integral curves to converge globally to the "stretched" path. We further introduce the extended dynamics to retain this appealing global convergence property for the desired path in the original lower-dimensional space. Both simulations and experiments are conducted to verify the theory.