ABSTRACT
Certainly, one of the most important notions in the theory of multifunctions is that of lower semicontinuity. Let us recall it. So, let X,Y be two topological spaces, https://www.w3.org/1998/Math/MathML"> F : X → 2 Y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq11538.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> a multifunction and https://www.w3.org/1998/Math/MathML"> x 0 ∈ X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq11539.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We say that F is lower semicontinuous at x 0 if, for every open set https://www.w3.org/1998/Math/MathML"> Ω ⊆ Y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq11540.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , the condition https://www.w3.org/1998/Math/MathML"> x 0 ∈ F - ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq11541.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> implies that https://www.w3.org/1998/Math/MathML"> x 0 ∈ i n t F - ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq11542.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , where https://www.w3.org/1998/Math/MathML"> F - ( Ω ) = { x ∈ X : F ( x ) ∩ Ω ≠ ∅ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429218576/b5164108-1897-4637-a93d-46e387408a02/content/eq11543.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We say that F is lower semicontinuous provided that it is so at each point of x.
