the category of T0 Alexandroff spaces is equivalent to the category of posets
Let be the category of all , Alexandroff spaces and continuous maps between them. Furthermore let be the category of all posets and order preserving maps.
Proof. Consider two functors:
such that , where is an induced partial order on an Alexandroff space and for continuous map. Analogously, let , where is an induced Alexandroff topology on a poset and for order preserving maps. One can easily show that and are well defined. Furthermore, it is easy to verify that equalities and hold, which completes the proof.
Remark. Of course every finite topological space is Alexandroff, thus we have very nice ,,interpretation” of finite spaces - finite posets (since functors and do not change set-theoretic properties of underlying sets such as finitness).
|Title||the category of T0 Alexandroff spaces is equivalent to the category of posets|
|Date of creation||2013-03-22 18:46:04|
|Last modified on||2013-03-22 18:46:04|
|Last modified by||joking (16130)|