Most CAD or other spatial data models, in particular boundary representation models, are called "topological" and represent spatial data by a structured collection of "topological primitives" like edges, vertices, faces, and volumes. These then represent spatial objects in geo-information- (GIS) or CAD systems or in building information models (BIM). Volume objects may then either be represented by their 2D boundary or by a dedicated 3D-element, the "solid". The latter may share common boundary elements with other solids, just as 2D-polygon topologies in GIS share common boundary edges. Despite the frequent reference to "topology" in publications on spatial modelling the formal link between mathematical topology and these "topological" models is hardly described in the literature. Such link, for example, cannot be established by the often cited nine-intersections model which is too elementary for that purpose. Mathematically, the link between spatial data and the modelled "real world" entities is established by a chain of "continuous functions" - a very important topological notion, yet often overlooked by spatial data modellers. This article investigates how spatial data can actually be considered topological spaces, how continuous functions between them are defined, and how CAD systems can make use of them. Having found examples of applications of continuity in CAD data models it turns out that of continuity has much practical relevance for CAD systems.
An example of topological spaces beyond @math in the context of relational databases is the theory of which converts a database schema into a topological space and tests it for so-called @math -cycles. This topological property severely affects the efficiency of some database query operations @cite_0 .
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I find a hierarchical arrangement of stocks traded in a financial market by investigating the daily time series of the logarithm of stock price. The topological space is a subdominant ultrametric space associated with a graph connecting the stocks of the portfolio analyzed. The graph is obtained starting from the matrix of correlation coefficient computed between all pairs of stocks of the portfolio by considering the synchronous time evolution of the difference of the logarithm of daily stock price. The hierarchical tree of the subdominant ultrametric space associated with the graph provides a meaningful economic taxonomy. For a long time topological relationships between spatial objects have been a focus of research in a number of disciplines like artificial intelligence, cognitive science, linguistics, robotics, and spatial reasoning. Especially as predicates they support the design of suitable query languages for spatial data retrieval and analysis in spatial databases and geographical information systems (GIS). Unfortunately, they have so far only been defined for and applicable to simplified abstractions of spatial objects like single points, continuous lines, and simple regions. With the introduction of complex spatial data types an issue arises regarding the design, definition, and number of topological relationships operating on these complex types. This article closes this gap and first introduces definitions of general and versatile spatial data types for complex points, complex lines, and complex regions. Based on the well known 9-intersection model, it then determines the complete sets of mutually exclusive topological relationships for all type combinations. Completeness and mutual exclusion are shown by a proof technique called proof-by-constraint-and-drawing. Due to the resulting large numbers of predicates and the difficulty of handling them, the user is provided with the concepts of topological cluster predicates and topological predicate groups, which permit one to reduce the number of predicates to be dealt with in a user-defined and or application-specific manner. We define a new topological summary for data that we call the persistence landscape. Since this summary lies in a vector space, it is easy to combine with tools from statistics and machine learning, in contrast to the standard topological summaries. Viewed as a random variable with values in a Banach space, this summary obeys a strong law of large numbers and a central limit theorem. We show how a number of standard statistical tests can be used for statistical inference using this summary. We also prove that this summary is stable and that it can be used to provide lower bounds for the bottleneck and Wasserstein distances. We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schroder and Simpson. We show the two constructions coincide when restricted to ω-continuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domain-theoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countably-based spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces.