![]() In this regard, experimental data from various independent sources ( WMAP, BOOMERanG, and Planck for example) interpreted within the standard family of metrics imply that the universe is flat to within only a 0.4% margin of error of the curvature density parameter. The shape of the universe remains a matter of debate in physical cosmology. Yet, in the case of simply connected spaces, flatness implies infinitude. For example, a multiply connected space may be flat and finite, as illustrated by the three-torus. Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one. For example, a universe with positive curvature is necessarily finite. There are certain logical connections among these properties. Connectivity: how the universe is put together as a manifold, i.e., a simply connected space or a multiply connected space.Flatness (zero curvature), hyperbolic (negative curvature), or spherical (positive curvature).Boundedness (whether the universe is finite or infinite).Its topological characterization remains an open problem. Several potential topological and geometric properties of the universe need to be identified. The main discussion in this context is whether the universe is finite, like the observable universe, or infinite. Assuming the cosmological principle, the observable universe is similar from all contemporary vantage points, which allows cosmologists to discuss properties of the entire universe with only information from studying their observable universe. Ĭosmologists distinguish between the observable universe and the entire universe, the former being a ball-shaped portion of the latter that can, in principle, be accessible by astronomical observations. The spatial topology cannot be determined from its curvature, due to the fact that there exist locally indistinguishable spaces that may be endowed with different topological invariants. The spatial curvature is described by general relativity, which describes how spacetime is curved due to the effect of gravity. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes general global properties of its shape as a continuous object. axiomatic set theory.The shape of the universe, in physical cosmology, is the local and global geometry of the universe. Perhaps more importantly, these definitions do not reflect modern mathematical thought on the topic 9, which is much more precise but also less accessible, e.g. However, it is worth noting that the descriptions used here are reflect the concepts generally used in GIS, and so are meant to be intuitive and not very formal. Since the basic concepts of set theory are used in this thesis in order to describe many other concepts, this section gives a very short primer using the same notation that is used in this thesis. While the study of set theory only formally started with Cantor, its intuitive and minimal concepts were later used in order to give a foundation to almost all areas of mathematics 8. Set theory is the branch of mathematics that studies sets, which are collections of abstract objects. §2.3 builds on these to present topology, which formalises notions such as the boundary and interior of an object or the relationships between multiple objects.Ģ.1 Elementary set theory and mathematical logic §2.2 introduces the basic concepts of geometry, which are used to describe the position, shape and orientation of objects. §2.1 introduces some concepts of elementary set theory and mathematical logic, which are later used in definitions in this thesis. The current chapter describes some of these formal notions and their relevant context, which are used to study the spatial modelling approaches presented in the upcoming chapters. ![]() These formal models make it possible to create and store digital representations of the world in a computer, and thus to use the power of a computer to easily solve spatial problems. In order to describe space unambiguously, people have thus turned to models that still describe geographical phenomena, but do so using formal notions derived from mathematics. However, these informal notions are error-prone and differ from person to person. Spatial modelling has its origins in the geographical notion of space, which is in turn based on our own observations of the world and empirical experience. ![]()
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