Why Does the Fixed Point Theorem Apply to Quines?
As Discrete Lizard points out, there exist more than one "fixed point theorem." The one you're mentioning makes sense in terms of topology and geometry, but you're right to realize that it does not apply to programming languages.

It turns out, though, that a more general notion of continuity, called Scott Continuity, can give you something that is applicable, namely the Kleene fixed-point theorem:


  Any Scott-continuous function over a directed-complete partial order has a least fixed point.


An earlier, and easier to understand, version of this is given by Knaster-Tarski:


  Any monotone function over a complete lattice has a fixed-point. In fact, the fixed points of such a function form a complete lattice.