Geospatial topology

Geospatial topology studies the rules concerning the relationships between the points, lines, and polygons that represent the features of a geographic region. For example, where two polygons represent adjacent counties, typical topological rules would require that the counties share a common boundary with no gaps and no overlaps. Similarly, it would be nonsense to allow two polygons representing lakes to overlap.

In spatial analysis the topological spatial relations are derived from the DE-9IM model, as spatial predicates about relations between points, lines, and/or areas: Equals, Contains, Covers, CoveredBy, Crosses, Disjoint, Intersects, Overlaps, Touches and Within. In network and graph representations the topology analysis is about topological objects such as faces, edges and nodes.

The ESRI White Paper GIS Topology explains that topology operations are used to manage shared geometry, define and enforce data integrity rules, support topological relationship queries and navigation, and build more complex shapes such as polygons, from primitive ones such as lines. A GIS for Educators worksheet at Linfiniti adds the detection and correction of digitising errors and carrying out network analysis. Topological error correction is explained in more detail in a paper by Ubeda and Egenhofer.

Geospatial analysis

Geospatial analysis, or just spatial analysis, is an approach to applying statistical analysis and other analytic techniques to data which has a geographical or spatial aspect. Such analysis would typically employ software capable of rendering maps processing spatial data, and applying analytical methods to terrestrial or geographic datasets, including the use of geographic information systems and geomatics.

Geographical information system usage

Geographic information systems (GIS), which is a large domain that provides a variety of capabilities designed to capture, store, manipulate, analyze, manage, and present all types of geographical data, and utilizes geospatial analysis in a variety of contexts, operations and applications.

Basic applications

Geospatial analysis, using GIS, was developed for problems in the environmental and life sciences, in particular ecology, geology and epidemiology. It has extended to almost all industries including defense, intelligence, utilities, Natural Resources (i.e. Oil and Gas, Forestry ... etc.), social sciences, medicine and Public Safety (i.e. emergency management and criminology), disaster risk reduction and management (DRRM), and climate change adaptation (CCA). Spatial statistics typically result primarily from observation rather than experimentation.

Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.

Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

Network topology

Network topology is the arrangement of the various elements (links, nodes, etc.) of a computer network. Essentially, it is the topological structure of a network and may be depicted physically or logically. Physical topology is the placement of the various components of a network, including device location and cable installation, while logical topology illustrates how data flows within a network, regardless of its physical design. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two networks, yet their topologies may be identical.

An example is a local area network (LAN): Any given node in the LAN has one or more physical links to other devices in the network; graphically mapping these links results in a geometric shape that can be used to describe the physical topology of the network. Conversely, mapping the data flow between the components determines the logical topology of the network.

Topology

There are two basic categories of network topologies: physical topologies and logical topologies.

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

Definition

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.