Atlas: Trivially and Locally Trivially $\cal C$ Maps by David Buhagiar

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International Conference on Topology and its Applications
August 23-27, 1999
Kanagawa University
Yokohama, Japan

Organizers
Yukinobu Yajima, the chairman, Masami Sakai, the vice-chairman, Yoshihiro Abe, Kazuhiro Sakai, Toshiji Terada, Kenichi Tamano, Akio Kato, Takao Hoshina, Hisao Kato, Kazuhiro Kawamura, Akira Koyama, Tsugunori Nogura

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Trivially and Locally Trivially C Maps
by
David Buhagiar
Mathematics Department, Okayama University, Okayama 700-8530
Coauthors: Takuo Miwa (Mathematics Department, Shimane University, Matsue 690-8504), Boris A. Pasynkov (Department of General Topology, Moscow State University, Moscow)

The study of General Topology is usually concerned with the category TOP of topological spaces as objects, and continuous maps as morphisms. It goes without saying that both of these concepts are equally important. Moreover, one can look at a space as a map from this space onto a singleton space and in this manner identify these two concepts. Bearing this in mind, a branch of General Topology which has become known as General Topology of Continuous Maps, or Fibrewise General Topology, was initiated. This field of research is concerned most of all in extending the main notions and results concerning topological spaces to that of continuous maps. For an arbitrary topological space Y one considers the category TOP_Y, the objects of which are continuous maps into the space Y, and for the objects f:X --> Y and g:Z --> Y, a morphism from f into g is a continuous map \lambda:X --> Z with the property f=g o \lambda.

In this work we give a possible systematic way of extending definitions from the category TOP to the category TOP_Y. This is done in the following way: Let C be some class of topological spaces. A map f:X --> Y is said to be trivally C ( \equiv TC) if it is parallel to a space C in C, i.e. there exists a space C in C and an embedding e:X --> Y×C such that f=pr_Y o e, where pr_Y:Y×C --> Y is the projection of the product onto the factor Y. A map f is said to be locally trivially C ( \equiv LTC) if for any y in Y, there exists a neighbourhood O_y of y such that the restriction f|_{f^-1O_y}:f^-1O_y --> O_y is a TC-map. One can note that in the definition of LTC-map, the space C_y in C can be different for every f|_{f^-1O_y}:f^-1O_y --> O_y.

We consider two classes of spaces as the collection C, the class of all metrizable spaces and the class of all linearly ordered topological spaces (i.e., LOTS). The authors have already introduced one possible way in defining a metrizable map, these maps are called MT-maps (metrizable type maps). The above mentioned method gives another possible way in defining a metrizable map, thus introducing the notion of TM-map (trivially metrizable) and its local version called LTM-maps (locally trivially metrizable). Examples are given to clarify the definitions and results.

Date received: June 30, 1999