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