Explore properties
| Identifier | Name | Description |
|---|---|---|
| P000000 | Priestley space | Compact Hausdorff space where for each pair of points x,y where x is not below y, there is a clopen upset U where x is in U and y is not in U. |
| P000001 | Esakia space | Compact Hausdorff Priestley space in which the downset of every clopen set is clopen. |
| P000002 | At most countable | The underlying set is finite or countably infinite. This concerns the whole construction, including its limit points, rather than the finite diagram window. |
| P000003 | Finite | The underlying set has finitely many points. Finite windows used to draw an infinite construction do not make this property true. |
| P000004 | Rooted | The order has a least element: a point below every point of the space or frame. |
| P000005 | Noetherian | The order has no infinite strictly ascending chain. A nonisolated least point may have transfinite depth without violating this condition. |
| P000006 | Finite width | There is a finite uniform bound on the cardinalities of antichains in the order. For a space or poset, it is assumed only locally: for each rooted upset. |
| P000007 | Greatest element | The order has a greatest element: a point above every point of the space or frame. |
| P000008 | Isolated root | The order has a least element and that point is isolated in the topology. |
| P000009 | Discrete | The topology is discrete (for posets with no specified topology, this is assumed). |
| P000010 | bi-Esakia space | Both the space and its order-dual are Esakia spaces. |
| P000011 | Finite depth | There is a uniform bound on the cardinality of the chains in the order. |
| P000012 | Clopen maximum | The maximum of the space is a clopen set. |
| P000013 | KM space | For every clopen subset, the maximum of the clopen is again clopen. |
| P000014 | Extremally order-disconnected | The upset of the closure of a union of clopen upsets is again a clopen upset. |
| P000015 | Extremally disconnected | Closure of a union of opens is open. |
| P000016 | Finite incomparability degree | For every point in the order, there are finitely many points which are incomparable to it. |
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