Topology
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Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct discipline developed, which was referred to in Latin as the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20^{th} century, topology had become an important area of study within mathematics.
The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse. For instance, the function y = x^{3} is a homeomorphism of the real line.
Topology includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division.
Knot theory studies mathematical knots.
See also: topology glossary for definitions of some of the terms used in topology and topological space for a more technical treatment of the subject.
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Topology topics
Some theorems in general topology
- Every closed interval in R of finite length is compact. More is true: In R^{n}, a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem).
- Every continuous image of a compact space is compact.
- Tychonoff's theorem: the (arbitrary) product of compact spaces is compact.
- A compact subspace of a Hausdorff space is closed.
- Every continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
- Every sequence of points in a compact metric space has a convergent subsequence.
- Every interval in R is connected.
- Every compact finite-dimensional manifold can be embedded in some Euclidean space R^{n}.
- The continuous image of a connected space is connected.
- A metric space is Hausdorff, also normal and paracompact.
- The metrization theorems provide necessary and sufficient conditions for a topology to come from a metric.
- The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.
- Any open subspace of a Baire space is itself a Baire space.
- The Baire category theorem: If X is a complete metric space or a locally compact Hausdorff space, then the interior of every union of countably many nowhere dense sets is empty.
- On a paracompact Hausdorff space every open cover admits a partition of unity subordinate to the cover.
- Every path-connected, locally path-connected and semi-locally simply connected space has a universal cover.
General topology also has some surprising connections to other areas of mathematics. For example:
- In number theory, Fürstenberg's proof of the infinitude of primes.
See also some counter-intuitive theorems, e.g. the Banach–Tarski one.
Some useful notions from algebraic topology
See also list of algebraic topology topics.
- Homology and cohomology: Betti numbers, Euler characteristic, degree of a continuous mapping.
- Operations: cup product, Massey product
- Intuitively attractive applications: Brouwer fixed-point theorem, Hairy ball theorem, Borsuk–Ulam theorem, Ham sandwich theorem.
- Homotopy groups (including the fundamental group).
- Chern classes, Stiefel–Whitney classes, Pontryagin classes.
Generalizations
Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
See also
- List of algebraic topology topics
- List of general topology topics
- List of geometric topology topics
- List of topology topics
- Publications in topology
- Topology glossary