We consider some questions concerning the nature and size of chains of open sets in Hausdorff spaces. The following results are obtained. Theorem 1. For every cardinal κ \kappa there exists a space X X in which all discrete subsets have cardinality at most κ \kappa and which contains a chain of ( 2 κ ) + {({2^\kappa })^ + } open sets . Theorem 2. If X X is regular and contains a chain of ( 2 κ ) + {({2^\kappa })^ + } open sets, then X × X X \times X contains a discrete subset of cardinality κ + {\kappa ^ + } . Theorem 3. Let M ( X ) M(X) denote the set of all maximal chains of open subsets of X X endowed with the Tychonoff topology . (i) | M ( X ) | ⩽ 2 w ( X ) \left | {M(X)} \right | \leqslant {2^{{\text {w}}(X)}} , and (ii) ψ ( M ( X ) ) ⩽ w ( X ) \psi (M(X)) \leqslant {\text {w}}(X) . Here w ( X ) {\text {w}}(X) denotes the weight of the space X X and ψ ( M ( X ) ) \psi (M(X)) denotes the pseudocharacter of the space M ( X ) M(X) .