On Some Special Classes of Continuous Maps
نویسندگان
چکیده
We present a survey on recent study of special continuous maps, like biquotient, triquotient, proper, perfect, open and étale maps and a selection of open problems in this area. 1. Special morphisms of Top Triquotient maps were introduced by E. Michael in [24] as those continuous maps f : X → Y for which there exists a map ( ) : OX → OY such that, for every U, V in the lattice OX of open subsets of X: (T1) U ♯ ⊆ f(U), (T2) X = Y , (T3) U ⊆ V ⇒ U ♯ ⊆ V , (T4) (∀y ∈ U ) (∀Σ ⊆ OX directed) f(y) ∩ U ⊆ ⋃ Σ ⇒ ∃S ∈ Σ : y ∈ S. It is easy to check that, if f : X → Y is an open surjection, then the direct image f( ) : OX → OY satisfies (T1)-(T4). If f : X → Y is a retraction, so that there exists a continuous map s : Y → X with f ◦ s = 1Y , then ( ) ♯ := s( ) satisfies (T1)-(T4). Moreover, if f : X → Y is a proper surjection (by proper map we mean a closed map with compact fibres: see [2]), then U ♯ := Y \ f(X \U) fulfills (T1)-(T4). That is, open surjections, retractions and proper surjections are triquotient maps. But there are triquotient maps which are neither of these maps (cf. [3, 15] for examples). However, we do not know whether these three classes of maps describe completely triquotient maps, in the sense we state now: Received August 24, 2005. The authors acknowledge partial financial assistance by Centro de Matemática da Universidade de Coimbra/FCT and by Unidade de Investigação e Desenvolvimento Matemática e Aplicações da Universidade de Aveiro/FCT. This text will be part of the book “Open Problems in Topology 2”, to be published by Elsevier BV.
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تاریخ انتشار 2005