# Topology (disambiguation)

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Topology , the study of surfaces, is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these properties are the topological invariants. In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

Topology may also refer to:

• Topology, the collection of open sets used to define a topological space
• Lawvere–Tierney topology of a topos
• Grothendieck topology of a category
• Topology (journal), a mathematical journal, with an emphasis on subject areas related to topology and geometry
• Spatial effects that cannot be described by topography, i.e., social, economical, spatial, or phenomenological interactions
• The specific orientation of transmembrane proteins
• Topology (electronics), a configuration of electronic components
• Network topology, configurations of computer networks
• Logical topology, the arrangement of devices on a computer network and how they communicate with one another

Logical topology is the arrangement of devices on a computer network and how they communicate with one another. Logical topologies describe how signals act on the network.

• Topology (musical ensemble), an Australian post-classical quintet
• Geospatial topology, the study or science of places with applications in earth science, geography, human geography, and geomorphology
• In geographic information systems and their data structures, topology and planar enforcement are the storing of a border line between two neighboring areas (and the border point between two connecting lines) only once. Thus, any rounding errors might move the border, but will not lead to gaps or overlaps between the areas.
• Also in cartography, a topological map is a greatly simplified map that preserves the mathematical topology while sacrificing scale and shape
• Topology is often confused with the geographic meaning of topography (originally the study of places). The confusion may be a factor in topographies having become confused with terrain or relief, such that they are essentially synonymous.

A geographic information system (GIS) is a system designed to capture, store, manipulate, analyze, manage, and present spatial or geographic data. GIS applications are tools that allow users to create interactive queries, analyze spatial information, edit data in maps, and present the results of all these operations. GIS sometimes refers to geographic information science (GIScience), the science underlying geographic concepts, applications, and systems.

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 cartography and geology, a topological map is a type of diagram that has been simplified so that only vital information remains and unnecessary detail has been removed. These maps lack scale, and distance and direction are subject to change and variation, but the relationship between points is maintained. A good example is the tube map of the London Underground or the map for the New York City Subway.

• In phylogenetics, the branching pattern of a phylogenetic tree
• TopologiLinux, a Linux distribution
• Topology (album), 1981 album by Joe McPhee

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, satisfying 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.

In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by William Lawvere (1971) and Myles Tierney.

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.

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Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus or Euclidean geometry. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

In mathematics, pointless topology is an approach to topology that avoids mentioning points. Topography is the study of the shape and features of land surfaces. The topography of an area could refer to the surface shapes and features themselves, or a description.

In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.

In topology, an Alexandrov topology is a topology in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open; in Alexandrov topologies the finite restriction is dropped.

In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.

Mathematics encompasses a growing variety and depth of subjects over history, and comprehension requires a system to categorize and organize the many subjects into more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve. In addition, as mathematics continues to be developed, these classification schemes must change as well to account for newly created areas or newly discovered links between different areas. Classification is made more difficult by some subjects, often the most active, which straddle the boundary between different areas.

Mathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena. Mathematical chemistry has also sometimes been called computer chemistry, but should not be confused with computational chemistry. A spatial network is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e. the nodes are located in a space equipped with a certain metric. The simplest mathematical realization is a lattice or a random geometric graph, where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches to that set to endow it with some additional meaning or significance. Spatial analysis or spatial statistics includes any of the formal techniques which study entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early development, using different analytic approaches and applied in fields as diverse as astronomy, with its studies of the placement of galaxies in the cosmos, to chip fabrication engineering, with its use of "place and route" algorithms to build complex wiring structures. In a more restricted sense, spatial analysis is the technique applied to structures at the human scale, most notably in the analysis of geographic data.

The following outline is provided as a topical overview of science:

Statistical geography is the study and practice of collecting, analysing and presenting data that has a geographic or areal dimension, such as census or demographics data. It uses techniques from spatial analysis, but also encompasses geographical activities such as the defining and naming of geographical regions for statistical purposes. For example, for the purposes of statistical geography, the Australian Bureau of Statistics uses the Australian Standard Geographical Classification, a hierarchical regionalisation that divides Australia up into states and territories, then statistical divisions, statistical subdivisions, statistical local areas, and finally census collection districts.

This is a glossary of terms that are or have been considered areas of study in mathematics.