Weihrauch Reducibility and Finite-Dimensional Subspaces

In this thesis we study several principles involving subspaces and decompositions of vector spaces, matroids, and graphs from the perspective of Weihrauch reducibility. We study the problem of decomposing a countable vector space or countable matroid into 1-dimensional subspaces. We also study the problem of producing a finite-dimensional or 1-dimensional subspace of a countable vector space, and related problems for producing finite-dimensional subspaces of a countable matroid. This extends work in the reverse mathematics setting by Downey, Hirschfeldt, Kach, Lempp, Mileti, and Montalb´an (2007) and recent work of Hirst and Mummert (2017). Finally, we study the problem of producing a nonempty subset of a countable graph that is equal to a finite union of connected components and the problem of producing a nonempty subset of a countable graph that is equal to a union of connected components that omits at least one connected component. This extends work of Gura, Hirst, and Mummert (2015). We briefly investigate some of these problems in the reverse mathematics setting