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1 2009

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Topologies and smooth maps on initial and final objects in the category of Frolicher spaces

Batubenge A., Tshilombo H.

Demonstratio Mathematica

2009

Vol. 42, nr 3

641-655

In this paper, we show that when the Frolicher smooth structure is induced on a subset or a quotient set, there are three natural topologies underlying the resulting object. We study these topologies and compare them in each case. It is known that the topology generated by strucure functions is the weakest one in which all functions and curves on the space are continuous. We show that on a subspace, it is rather the trace topology which has this property, while the three topologies are coincident on the quotient space. We construct a base for the Frolicher topology and using either a base or a subbase in the sense of A. Frolicher [9], we characterise the morphisms of this category.