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