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The purpose of this thesis is to explore the spatial outcomes of the extension of coal mining onto the concealed coalfield of east Durham in the first half of the nineteenth century. Here, in contrast to the long-established exposed sections of the North-Eastern coalfield, raining was developed suddenly consequent upon the first successful sinkings through the Magnesian Limestone at Hetton-le-Hole between l820 and 1822.In more detail, the first objective of the work is to reconstruct the agrarian base upon which the colliery landscape was superimposed. The agrarian base is presented as a cadaster composed of patterns of landownership, landholding, settlements, fields and land-use that provided a spatial matrix within which the process of mining colonisation developed. Subsequently, the following three questions are examined: (i) to what extent did the legal relationships between the east Durham landowners and the colliery companies structure the emergent locational pattern of the colliery landscape in terms of the siting of the mines with their associated surface installations, the colliery settlements and the transport lines? (ii) what was the social structure of these rapidly-developed mining communities? (iii) what effect did the sudden creation of large-scale centres of mining employment have upon patterns of labour mobility? By this means, it is intended to analyse, in terms of a regional case study, the impact of coal mining upon the human geography of east Durham. Because of the nature of these objectives, the work is essentially ideographic rather than nomothetic in concept, empirical rather than theoretical in approach, the overall aim being to present the spatial outcomes of the complex and at times subtle relationships between man, technology and' resource in this small corner of the Industrial Revolution

This thesis explores families of metric spaces. It has two parts. First, the Kuratowski embedding is an isometric embedding of a metric space, M, into the space of bound, Borel functions over M, equipped with the sup norm. We extend this map to a family of maps by averaging over metric r--balls. The image of M under this map can be regarded as a deformation. After restricting our metric spaces to Riemannian manifolds, we explore how curvature affects this deformation. Furthermore, we give a complete description of the deformation of the n-sphere. Second, we prove a diffeomorphsim stability theorem. The smallest r so that a metric r--ball covers a metric space M is called the radius of M. The volume of a metric r-ball in the space form of constant curvature k is an upper bound for the volume of any Riemannian manifold with sectional curvature ≥ k and radius ≤ r. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.