Teichmuller spaces for surfaces of infinite type can be studied with geometric techniques, like the pair of pants decomposition and and Fenchel-Nielsen coordinates. In this case we have countably infinitely many pairs-of-pants and coordinates. With these techniques we can describe, under certain hypotheses, the topology given by the Teichmuller distance and the topology given by the "length spectrum" distance.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3050