Techniques from Set Theory and Logic have been used in Topology for
a long time. In the last decade or so, topologists have been explicitly
exploiting the fact that many topological properties are not first-order.
I presented an
elementary exposition of the technique of elementary submodels
of the universe in Galway in 1997. A thorough introduction by A. Dow
can be found in Topology Proceedings, volume 13 (1988), no 1, pages 17--72.