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Title
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Frame theoretic methods in topology and analysis
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Author
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Abstract
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| The study of frames is also known as pointfree topology. As the name suggests, this is a way of studying spaces without (mentioning) points. This idea is more natural than one might initially think: when drawing a point on paper, we do not draw an actual point, but rather a “spot”, which can be reduced in size if that would be required to serve our purposes. In topology, spots of varying sizes are modelled by open sets, which become the building blocks of so-called frames. In many cases, it is possible to reverse this process by defining points as being frame homomorphisms to 2={0,1}. Approach spaces on the other hand can be used to combine the advantages of both topological and metric spaces. Topological spaces behave better with respect to metric spaces in the sense that we can take arbitrary products of topological spaces which still behave well. On the other hand, topological spaces are more of the all-or-nothing-type. Two points can be either the same, or not. A point can be inside a subspace, or not. Here, metric spaces have the advantage: we can say exactly how far two points are apart from each other. Approach spaces remedy this by defining the distances between a point and a set, not between points. One of the characterizations of approach spaces is the lower regular function frame L, consisting of contractions to P=[0,∞], which is in fact a frame. In this thesis, we first remark that whereas point-set distances and lower regular function frames L have been extensively studied, their ‘interior’ counterparts ι and U were left mostly ignored. We begin by defining and studying ι and also consider upper regular function frame U. We then consider another property of approach spaces which was not extensively studied yet, namely normality. Normality is an important tool in order to prove existence of extensions of function, such as Urysohn maps, Tietze's extension and Katetov-Tong's insertion. We then remark that, as for contractions, the set of semicontinuous maps to P also defines a frame. Moreover, a frame of opens can be described equivalently by the set of continuous maps to 2. Given that the points of a frame are defined by the frame homomorphisms to 2, we can replace both instances of 2 by another space/frame F. We investigate the transitions between spaces and F-frames and find a connection with free distributive lattices. |
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Language
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English
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Publication
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Antwerp
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University of Antwerp, Faculty of Science, Department of Mathematics
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2021
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Volume/pages
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xiv, 121 p.
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Note
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Lowen, Wendy [Supervisor]
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Sioen, Mark [Supervisor]
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Full text (publisher's version - intranet only)
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