Spaces of Mappings on Topological Products with Applications to Homotopy Theory

On the Homotopy Classification of the Extensions of a Fixed Map

“…Then the natural function 0: X<-KXT)-+(XK)T, given by (*/) (0 (*)=/(*,<), kEK,tET, is a homeomorphism if P is a CW complex such that KXT, given the product topology, is also a CW complex (the proof is elementary; cf. [2] and [9] for other cases in which 0 is a homeomorphism).…”

Quantum families of maps

“…The next theorem extends this to spaces of continuous mappings of topological spaces. Theorem 3.1 (J.R. Jackson [4]). Let X, Y and E be topological spaces such that X is Hausdorff and…”

Function Spaces and Product Topologies

“…In this section X 7 will mean the space of continuous functions Y -*• X with the compact-open topology (8); X T is a Hausdorff space [(8) Theorem 7.4].…”

“…Let M(C,U) = {ueX z :u(C) £ U) be a sub-basic set for the compact-open topology on X z . Then r l {M{C, U)) = {ugeX 7 :us X z , u(C) c U} = {ugeX T :ueX z , ugg-^C) £ U} = {veH:vg-HC) £ U}, which is a sub-basic set for the compact-open topology on H. Thus j is continuous, and A is a homeomorphism into.…”