Pseudofinite and pseudocompact metric structures
  We initiate the study of pseudofiniteness in continuous logic. We introduce a related concept, namely that of pseudocompactness, and investigate the relationship between the two concepts. We establish some basic properties of pseudofiniteness and pseudocompactness and provide many examples. We also investigate the injective-surjective phenomenon for definable endofunctions in pseudofinite structures. 
Isaac Goldbring and Vinicius Cifu Lopes
math.LO