# Set Functions

A continuum is a non-empty compact connected metric space. Given a continuum we define the hyperspaces:

2{X} = { A \subset X : A is closed and nonempty} C(X) = { A \in 2{X} : A is connected}

2{X} is called the hyperspace of closed nonempty subsets of X and C(X) is called the hyperspcae of subcontinua of X. The hyperspace 2{X} is topologized with the Haussdorf metric.

Given a continuum, the set functions T:2{X} \to 2{X} and K:2{X} \to 2{X} are functions defined on the hyperspace 2{X} of the continuum X in the following way, for every A in 2{X}:

T(A) = {x \in X : if W is a continuum containing x in its interior, then A \cap W = \nonempty }

K(A) = \cap {W \in C(X) : A \subset Int(W) }

Examples and properties of these set functions are going to be presented.

This talk is part of the Analysis Seminar series.