### The Kuratowski Closure-Complement Problem

It is possible to construct a Kuratowski 14‑set by selecting an appropriate combination of subsets from the list below. The rationals and irrationals in the open interval (2, 4) are denoted by Q(2, 4) and I(2, 4), respectively. Operations are applied to the seed set E in right-to-left order. For example, ckE denotes the complement of the closure of E, where c denotes complement and k denotes closure.

The Hasse diagram below shows all set containments that hold in general among the 14 possible sets, with larger sets appearing above smaller sets. The full collection of distinct sets generated by E is displayed in the shaded entries. These entries also appear in a different order in a table at the bottom of the page, to help show that their number is maximal.

 kE kcE ∅ kckckE E kckckcE Count: 2 ckckE kckcE ckckcE kckE (–∞, ∞) ckckckcE cE ckckckE ckcE ckE
 1 2 4 5 (–∞, 1) {1} (1, 2) {2} Q(2, 4) I(2, 4) {4} (4, 5) {5} (5, ∞)
 Count: 2 E ∅ cE (–∞, ∞) kE E kcE cE ckE ckcE kckE kckcE ckckE ckckcE kckckE kckckcE ckckckE ckckckcE kckckckE kckE kckckckcE kckcE

The two bottom entries above remain constant because kckckckE = kckE for all sets E.