Ferroelectric topological objects provide a fertile ground for exploring emerging physical properties that could potentially be utilized in future nanoelectronic devices. Here, we demonstrate quasi-one-dimensional metallic high conduction channels associated with the topological cores of quadrant vortex domain and center domain (monopole-like) states confined in high quality BiFeO3 nanoislands, abbreviated as the vortex core and the center core. We unveil via the phase-field simulation that the superfine metallic conduction channels along the center cores arise from the screening charge carriers confined at the core region, whereas the high conductance of vortex cores results from a field-induced twisted state. These conducting channels can be reversibly created and deleted by manipulating the two topological states via electric field, leading to an apparent electroresistance effect with an on/off ratio higher than 103. These results open up the possibility of utilizing these functional one-dimensional topological objects in high-density nanoelectronic devices, e.g. nonvolatile memory.