Induced mappings on the hyperspace of totally disconnected sets

Abstract The symbol TD ( X ) denotes the hyperspace of all nonempty totally disconnected compact subsets of a Hausdorff space X . This hyperspace is endowed with the Vietoris topology. For a mapping between Hausdorff spaces f : X → Y , define the induced mapping TD ( f ) : TD ( X ) → TD ( Y ) by TD ( f )( A ) = f ( A ) (the image of A under f ). In the current paper, we study the relationships between the condition f belongs to a class of mappings between Hausdorff spaces 𝕄 and the condition TD ( f ) belongs to 𝕄.