Department of Mathematics
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I will show parts of a standard proof from algebra showing the equivalence of equational compactness and pure-injectivity. In particular, I will show that if N is a "pp compact" module, then N is injective over pure embeddings. The module N is pp compact if, whenever Σ is a set of pp formulas with parameters from N (in arbitrarily many variables) such that every finite subset of Σ has a solution in N, then Σ itself has a solution in N.

I will then explain one attempt to develop a parallel proof for topological modules. In this version, both "pp compact" and "pure-injective" are taken to refer to topological positive primitive formulas with both individual variables and set variables free.