We study topological versions of paths, cycles and spanning trees in infinite graphs with ends that allow more comprehensive generalizations of finite results than their standard notions. For some graphs it turns out that best results are obtained not for the standard space consisting of the graph and all its ends, but for one where only some ends are added as new points, while rays from other ends are made to converge to certain vertices.
See Section 7 for problems related to end-faithful spanning trees.
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