  We show in ZF that: (i) Every subcompact metrizable space is completely
metrizable, and every completely metrizable space is countably subcompact. (ii)
A metrizable space X=(X,T) is countably compact iff it is countably subcompact
relative to T. (iii) For every metric space X=(X,d), X is compact iff it is
subcompact relative to T. We also show: (iv) The negation of each of the
statements, (a) every countably subcompact metrizable space is completely
metrizable, (b) every countably subcompact metrizable space is subcompact, (c)
every complete metrizable space is subcompact is relatively consistent with ZF.
