Representing quasi-orders by embeddability ordering of families of topological spaces

An elementary argument constructs, for each cardinal alpha, a topological s pace whose subspaces, ordered by homeomorphic embeddability, can model ever y partial order on alpha-many points. We show how to modify this procedure to deal also with quasi-orders (where the antisymmetry condition may fail), obtaining an initial estimate of the cardinality of the space then require d.