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Regarding III, the Alexandrov-Urysohn Theorem gives sufficient conditions.
Any zero-dimensional, separable, nowhere compact, and completely metrizable space is homeomorphic to J$J$.
Any zero-dimensional, separable, nowhere compact, and completely metrizable space is homeomorphic to J.
Any zero-dimensional, separable, nowhere compact, and completely metrizable space is homeomorphic to $J$.
Any zero-dimensional, separable, nowhere compact, and complete metriccompletely metrizable space is homeomorphic to JJ.
Any zero-dimensional, separable, nowhere compact, and complete metric space is homeomorphic to J.