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This is joint work with Carl Mummert.  We initiate the reverse
mathematics of general topology.  We show that a certain metrization
theorem is equivalent to $\Pi^1_2$ comprehension.  An $MF$ space is
defined to be a topological space of the form $MF(P)$ with topology
generated by ${ N_p | p \in P }$.  Here P is a poset, $MF(P)$ is the
set of maximal filters on $P$, and $N_p = { F \in MF(P) | p \in F }$.
If the poset $P$ is countable, the space $MF(P)$ is said to be
countably based.  The class of countably based $MF$ spaces can be
defined and discussed within the subsystem $ACA_0$ of second order
arithmetic.  One can prove within $ACA_0$ that every complete
separable metric space is homeomorphic to a countably based $MF$ space
which is regular.  We show that the converse statement, ``every
countably based $MF$ space which is regular is homeomorphic to a
complete separable metric space," is equivalent to $\Pi^1_2-CA_0$.
The equivalence is proved in the weaker system $\Pi^1_1-CA_0$.  This
is the first example of a theorem of core mathematics which is
provable in second order arithmetic and implies $\Pi^1_2$
comprehension.
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