On the continuity of functors of the type C(X, Y)
Keywords:
function space, functor C(X,Y), continuous functor, inverse spectrum, direct spectrumAbstract
We consider the category P, the objects of which are pairs of topological spaces (X, Y). Each such pair (X, Y) is assigned the space of continuous maps Cτ(X, Y) with some topology τ. By imposing some restrictions on objects and morphisms of category P, we define a subcategory K ⊂ P, for which the above map is a functor from K to the category Top of topological spaces and continuous maps. The following question is investigated. What are the additional conditions on K, under which the above functor is continuous? Along the way the problem of finding the limit of the inverse spectrum in the category P is solved. We show, that it reduces to finding the limits of the corresponding direct spectrum and inverse spectrum in the category Top. Point convergence topology, compact-open topology and graph topology are considered as the topology τ.
References
- Kukrak HO, Timokhovich VL, Frolova DS. [Some topological properties of the functor of C(X, Y)]. Trudy Instituta matematiki NAN Belarusi. 2018;26(1):71–78. Russian.
- Kukrak HO, Timokhovich VL. [Some topological properties of mapping spaces]. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2010;1:144–149. Russian.
- Timokhovich VL, Frolova DS. [On infimal topology of mapping spaces]. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2011;2:136–140. Russian.
- Timokhovich VL, Frolova DS. [Infimal topology of mapping spaces and evaluation map]. Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika. 2012;1:68–72. Russian.
- Timokhovich VL, Frolova DS. [Topologies of uniform convergence. The property in the sense of Arens – Dugundji and the sequential property]. Izvestiya vuzov. Matematika. 2013;9:45–58. Russian.
- Engelking R. General topology. Berkeley: John L. Keller; 1955. 298 p. Russian edition: Engelking R. Obshchaya topologiya. Moscow: Mir; 1986. 752 p.
- Bacon P. The compactness of countably compact spaces. Pacific Journal of Mathematics. 1970;32(3):587–592.
- Naimpally S. Graph topology for function spaces. Transactions of the American Mathematical Society. 1966;123:267–272.
- Fedorchuk VV, Filippov VV. Obshchaya topologiya. Osnovnie konstruktsii [General topology. The main constructions]. Moscow: Fizmatlit; 2006. 336 p. Russian.
- Bucur I, Deleanu A. Introduction to the theory of categories and functors. New York: NYWiley; 1968. 224 р. Russian edition: Bucur I, Deleanu A. Vvedenie v teoriyu kategoriy i funktorov. Raikova DA, Retakh VS, translators. Moscow: Mir; 1972. 259 p.
- Aleksandrian RA, Mirzakhanyan EA. Obshchaya topologiya [General topology]. Moscow: Vyshaya shkola; 1979. 336 p. Russian.
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