In his survey article in the Handbook of Set-theoretic Topology on cardinal characteristics of the continuum and small cardinals arising in topology, van Douwen introduced three such invariants of a separable metrizable space, M, namely cof(K(M)), kc(M) and k(M). Each invariant asks for the minimum size of a family of compact subsets of M with certain properties.The third invariant, k(M), requires that the compact subsets witness the k-space property of M. In this talk we aim to understand not just the size, but the “shape” of compact families witnessing the k-space property (k-structures), and the “shape” of families of convergent sequences witnessing sequentiality (sequential structures), of a separable metrizable space.

Our primary tool will be an extension, due to Vojtas, of the Tukey order on directed sets to general relations. A natural question arising from this work will have as its answer, `the omega_1 st fixed point of the aleph function’.