This dissertation discusses several set-valued set functions on compact Hausdorff spaces. The definitions of the set functions are similar to that of the well-known set function T. In addition to set functions T and K, three new set functions will be discussed.

Let S be a compact Hausdorff space, $[/cal F]$(S) be the collection of nonempty closed subsets of S and $[/cal C] /subset[/cal F](S).$ If $A/subset S$ then $/Theta(A)$ is the complement of the set of points x in S for which $x/in[/rm int]F$ for some $F/in[/cal C]$ which misses A. From this definition, many set functions can be defined by specifying $[/cal C].$ The set function H is defined using $[/cal C]=/[U/in[/cal F](S):$ each component of U has empty interior$/].$ J is defined using $[/cal C]=/[U/in[/cal F](S):$ intU contains a subcontinuum with nonempty interior$/],$ and $/Delta$ is defined using $[/cal C]=/[U/in [/cal F](S):/ U$ is dense in $U/],$ where $/ U=/cup/[W:W$ is a subcontinuum of $S,W/subset U$ and int$W/not= /emptyset/].$

Chapter 1 provides some background information. Chapter 2 discusses the basic properties of an arbitrary set function $/Theta$ and provides additional results for specific functions. Some interesting results include the following:

(1) K is the identity set function if and only if S is colocally connected.

(2) If S is any compact Hausdorff space, then $H(A)=A/cup H(/emptyset).$

(3) J is the identity set function if and only if S is almost connected im Kleinen.

(4) For $A/in[/cal F](S),/ J(A)/subset/Delta(A)/subset T(A).$

Chapter 3 discusses the relationships between $/Theta/sb[s]f/sp[-1](A)$ and $f/sp[-1]/Theta/sb[Z](A)$ where f is a continuous mapping from one compact Hausdorff space onto another and $/Theta/sb[S]$ and $/Theta/sb[Z]$ represent $/Theta$ on the specified space. The results of this chapter can be applied to show that certain types of maps cannot exist. For instance, there cannot exist a continuous open map from any almost connected im Kleinen space onto an indecomposable continuum.

Chapter 4 discusses the unions, intersections and compositions of the set functions and Chapter 5 discusses the behavior of $/Theta(/emptyset).$ Some interesting results from Chapters 4 and 5 are as follows:

(1) For all $A/in[/cal F](S),/ (H/cap J)(A)=A.$

(2) For all $A/in[/cal F](S),/ H/circ/Theta(A)/subset/Theta/circ H(A).$

(3) S is almost connected im Kleinen if and only if $J(A)=A/cup J(/emptyset).$