Semilattice structures on certain non-metric continua



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An arc is a compact, connected, totally ordered topological space. In this dissertation, the codimension and cohomology of certain subsets of a Cartesian product of two arcs are computed. In addition, it is proved that an arc P contained in the Cartesian product of two arcs A and B separates the product space A x B if the endpoints of P separate the boundary of A x B. The metric version of this theorem was proved by G. T. Whyburn. The lattice product of two arcs is the Cartesian product of the arcs with the induced lattice operations. Topological lattices whose underlying spaces are homeomorphic to the Cartesian product of two arcs are studied. It is shown that each such lattice is distributive and can be topologically and algebraically embedded into the lattice product of two arcs. Conditions are given where the embedding is onto the lattice product. These results are extensions of the metric versions, which were proved by Wallace in 1958, and Clark and Eberhart in 1968. Next, the question of the existence of topological semi lattice structures with identities on trees is considered. A tree is a compact, connected, Hausdorff topological space in which each pair of points is separated by a third point. In 1964, Koch and McAuley proved that each metric tree supports the structure of a topological semi lattice with identity. Then, in 1966, Eberhart gave an example of a non-metrizable, non-first-countable tree which does not support such a structure. In this dissertation, an example of a first-countable tree not supporting such a structure is given. Various conditions are stated under which a tree supports the structure of a topological semi lattice with identity. One such condition is that T is a tree which contains a maximal arc [a,b] such that each component of T[a,b] is separable. It is shown that this condition does not force T to be metrizable.