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**The ABC of order and topology.**
*(English)*
Zbl 0735.18005

Category theory at work, Proc. Workshop, Bremen/Ger. 1991, Res. Expo. Math. 18, 57-83 (1991).

Summary: [For the entire collection see Zbl 0732.00006.]

The classical equivalence between quasiordered sets and Alexandrov- discrete spaces (\(A\)-spaces) extends to one-to-one correspondences between certain categories of generalized ordered sets where the order relations are still idempotent (hence transitive) but not necessarily reflexive, and categories of topological spaces whose topologies have a smallest base (\(B\)-spaces) or are completely distributive lattices (\(C\)- spaces). We introduce the appropriate types of morphisms making these one-to-one correspondences functorial and extend the results to arbitrary closure spaces which need not be topological. Moreover, we study various reflections, adjoint situations, equivalences and dualities between certain categories whose objects carry an order-theoretical or a topological structure. Our approach will provide the common framework for old and new topological representations of certain types of lattices by suitable spaces, and vice versa. The categorical point of view makes the manifold links between ordered and topological structures more transparent and yields the appropriate background for applications of topology to order theory, and conversely.

The classical equivalence between quasiordered sets and Alexandrov- discrete spaces (\(A\)-spaces) extends to one-to-one correspondences between certain categories of generalized ordered sets where the order relations are still idempotent (hence transitive) but not necessarily reflexive, and categories of topological spaces whose topologies have a smallest base (\(B\)-spaces) or are completely distributive lattices (\(C\)- spaces). We introduce the appropriate types of morphisms making these one-to-one correspondences functorial and extend the results to arbitrary closure spaces which need not be topological. Moreover, we study various reflections, adjoint situations, equivalences and dualities between certain categories whose objects carry an order-theoretical or a topological structure. Our approach will provide the common framework for old and new topological representations of certain types of lattices by suitable spaces, and vice versa. The categorical point of view makes the manifold links between ordered and topological structures more transparent and yields the appropriate background for applications of topology to order theory, and conversely.

### MSC:

18B30 | Categories of topological spaces and continuous mappings (MSC2010) |

06F30 | Ordered topological structures |

54B30 | Categorical methods in general topology |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |