Fibrewise retraction and extension
by
Takuo Miwa
Department of Mathematics, Shimane University, Matsue, Japan

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 spaceonto 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 B one considers the category TOP_B, the objects of which are continuous maps into the spaceB, and for the objects f:X --> B and g:Y --> B, a morphism from f into g is a continuous map \lambda:X --> Z with the property f=g o \lambda.

Retract theory is one of the most important in topology. In this work, we study fibrewise retracts and extensions. We introduce notions of absolute (nbd) retracts (or extensor) over B relative to a fibrewise class Q_B, and a notion of fibrewise adjunction spaces. We give some results about the relations of fibrewise ANR and ANE and fibrewise contractibility.

Date received: July 11, 1999