Let \(X = \prod_{\alpha < \kappa} X_\alpha\) be the product of topological spaces \(X_\alpha\). A subset \(D \subset X\) is called thin if any two distinct members of \(D\) are distinct in at least two coordinates. \(D\) is very thin if any two distinct members are distinct in all coordinates. \(D\) is said to be slim if for every nonempty proper subset \(K \subset \kappa\) and every \(v \in \prod_{\alpha \in K} X_\alpha\), the set \(D \cap C (v)\) is nowhere dense in \(C(v)\) where \(C(v) = \{ x \in X : x \upharpoonright K = v \}\) is the cross-section of \(X\) at \(v\). Clearly, very thin implies thin, and for products of two spaces the notions are equivalent. Also, if all \(X_\alpha\) are dense-in-themselves \(T_1\)-spaces, then every very thin set is slim.
The authors prove a number of results on the existence and non-existence of dense sets with one of these three properties. For example, the infinite power of any space has a thin dense set. An infinite product of separable dense-in-themselves Hausdorff spaces has a slim and thin dense set. A product of metrizable dense-in-themselves spaces has a slim dense set. However, there is a metrizable dense-in-itself space whose square has no thin dense set. On the other hand, under the continuum hypothesis CH there is a countable regular space \(X\) such that \(X^3\) has a thin dense subset but no slim (and hence no very thin) dense subset. Thus the notions of thin and slim are incomparable.