On continuity of group-valued mappings with respect to Chabauty topology

Let 𝐺 be a locally compact group and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">S</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">U</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> SUB(G) be the set of closed subgroups of 𝐺 equipped with the Chabauty topology. In this paper, we establish some necessary and sufficient conditions for a map <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="bold">T</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo stretchy="false">→</m:mo> <m:mrow> <m:mi mathvariant="script">S</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">U</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> T X(G) , from a topological space 𝑋 into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="script">S</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">U</m:mi> <m:mo>⁢</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> SUB(G) , to be continuous.