  We define a moduli space of translation structures on the open topological
disk with a basepoint and endow it with a locally-compact metrizable topology.
We call this the immersive topology, because it is defined using the concept of
immersions: continuous maps between subsets of translation surfaces that
respect the basepoints and the translation structures. Immersions induce a
partial ordering on the moduli space, and we prove the ordering is nearly a
complete lattice in the sense of order theory: The space is only missing a
minimal element. Subsequent articles will uncover more structure and develop a
topology on the space of all translation structures.
