Abstract
If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2κ. Our main result is a vivid construction of 2κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.
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