The Vietoris topology F on the hyperspace of non-empty closed^subsets of a space X is the supremum of the upper semi-finite (U)^and lower semi-finite (L) topologies on the hyperspace. This^hyperspace with the U topology is denoted U(X) and the hyper-^space of U(X) with the L topology is denoted L(U)(X)). The space^L(U(X)) is at T(,0) space which contains as subsets many different^extensions of X. In particular, if X is Tychonoff, the Stone-Cech(' )^compactification (beta)X is one such subset; this relies on the z-^ultrafilter description of (beta)X. As a subset of L(U(X)), (beta)X is compactbut not closed. This fact allows us to represent each Hausdorffcompactification of X as the union of X (L-HOOK) (beta)X and certain points ofthe boundary of (beta)X. More generally, by altering the U topology, wecan do the same for each Wallman compactification of X.

The second part of the thesis generalizes ultrafilters in the following way: we call a collection of closed subsets of a T(,4) space X an n-linked system if and only if every subcollection of (LESSTHEQ) n sets has non-empty intersection in X. The set (SUMM)(,n)(X) of all maximal n-linked systems is a subset of L(U(X)); it is an extension of X in which X is not a dense subset. The space (SUMM)(,2)(X) was discovered by J. de Groot and is called the superextension of X. It is a supercompact Hausdorff space, and has several nice properties. For n (GREATERTHEQ) 3, (SUMM)(,n)(X) has similar, but not so well-behaved, properties. It is not compact, but is cocompact, hence a Baire space. If X is compact metric, then (SUMM)(,n)(X) is metrizable, and conversely. For n = 2, (SUMM)(,2)(X) is connected and locally connected if and only if X is connected. Connectivity for n (GREATERTHEQ) 3 is more difficult to prove. For example, we have (SUMM)(,n)(X) connected if X is a continuum which satisfies any one of the properties: (1) no finite subset separates X, (2) X is linearly ordered, (3) X is arcwise connected and metric. For each j (ELEM) {2,...,n-1} there is a topology L(,j) for (SUMM)(,n)(X) which is weaker than the subspace topology L; there is also a set (SUMM)(,n)('j)(X) derived from (SUMM)(,n)(X) such that ((SUMM)(,n)('j)(X),L) is a continuous open image of ((SUMM)(,n)(X),L(,j-1)) (we take L(,1) = L).

Finally, we examine the U, L, and F topologies for the hyperspaces of an arbitrary space X. Some very weak separation properties for X characterize certain properties of the closure operator in U(X) (L(X),F(X)) and determine when each space contains a homeomorphic copy of X.