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In mathematics, particularly in functional analysis and topology, closed graph is a property of functions.[1][2] A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph.[3]
This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.
- ^ Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.
- ^ Ursescu, Corneliu (1975). "Multifunctions with convex closed graph". Czechoslovak Mathematical Journal. 25 (3): 438–441. doi:10.21136/CMJ.1975.101337. ISSN 0011-4642.
- ^ Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.
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