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We construct a continuum of non-homeomorphic compact subspaces of the realline R without singleton components. Thus from the purely topological point ofview the real line contains not only more closed sets than open sets but alsomore closures of open sets than open sets. On the other hand, we show that thisdiscrepancy vanishes either if the topological point of view is sharpened inthe metrical or in the order-theoretical direction, or if R is replaced withR^n for dimension n>1. Furthermore, we track down a continuum of topologicaltypes of closed and totally disconnected subsets of R. In doing so we alsotrack down a continuum of metrical types of infinite, discrete subsets of theunit interval [0,1]. (As a consequence, any countably infinite discrete spacehas a continuum of non-homeomorphic metrizable compactifications.)

On the variety of Euclidean point sets