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

An elementary argument constructs, for each cardinal $\alpha$, a topological space whose subspaces, ordered by homeomorphic embeddability, can model every 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 required.