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Abstract

A group is left orderable if there exists a strict total ordering of its elements that is invariant under multiplication from the left. The set of all left orderings of a group comes equipped with a natural topological structure and group action, and is called the space of left orderings. This thesis investigates the topology of the space of left orderings for a given group, by analyzing those left orderings that correspond to isolated points and by characterizing the orbits of the natural group action using maps from a certain free object. Lastly, we find an application of the space of left orderings in the field of 3-manifold topology, by using compactness to show that certain fundamental groups are not left orderable.