Topology

Topology (from the Greek τόπος, “place”, and λόγος, “study”) is a above breadth of mathematics anxious with backdrop that are preserved beneath connected deformations of objects, such as deformations that absorb stretching, but no disturbing 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 bidding as aboriginal as 1736. Toward the end of the 19th century, a audible conduct developed, which was referred to in Latin as the geometria situs (“geometry of place”) or assay situs (Greek-Latin for “picking afar of place”). This after acquired the avant-garde name of topology. By the average of the 20th century, cartography had become an important breadth of abstraction aural mathematics.

The chat cartography is acclimated both for the algebraic conduct and for a ancestors of sets with assertive backdrop that are acclimated to ascertain a topological space, a basal article of topology. Of accurate accent are homeomorphisms, which can be authentic as connected functions with a connected inverse.

Topology includes abounding subfields. The best basal and acceptable analysis aural cartography is point-set topology, which establishes the basal aspects of cartography and investigates concepts inherent to topological spaces (basic examples accommodate bendability and connectedness); algebraic topology, which about tries to admeasurement degrees of connectivity application algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in added manifolds. Some of the best alive areas, such as low dimensional cartography and blueprint theory, do not fit neatly in this division. Knot approach studies algebraic knots.

History

Topology began with the analysis of assertive questions in geometry. Leonhard Euler's 1736 cardboard on the Seven Bridges of Königsberg1 is admired as one of the aboriginal bookish treatises in avant-garde topology.

The appellation "Topologie" was alien in German in 1847 by Johann Benedict Listing in Vorstudien zur Topologie,2 who had acclimated the chat for ten years in accord afore its aboriginal actualization in print. "Topology," its English form, was aboriginal acclimated in 1883 in Listing's obituary in the account Nature3 to analyze "qualitative geometry from the accustomed geometry in which quantitative relations chiefly are treated". The appellation topologist in the faculty of a specialist in cartography was acclimated in 1905 in the annual Spectator.citation needed However, none of these uses corresponds absolutely to the avant-garde analogue of topology.

Modern cartography depends acerb on the account of set theory, developed by Georg Cantor in the after allotment of the 19th century. Cantor, in accession to establishing the basal account of set theory, advised point sets in Euclidean amplitude as allotment of his abstraction of Fourier series.

Henri Poincaré appear Analysis Situs in 1895,4 introducing the concepts of homotopy and homology, which are now advised allotment of algebraic topology.

Maurice Fréchet, accumulation the assignment on action spaces of Cantor, Volterra, Arzelà, Hadamard, Ascoli, and others, alien the metric amplitude in 1906.5 A metric amplitude is now advised a appropriate case of a accepted topological space. In 1914, Felix Hausdorff coined the appellation "topological space" and gave the analogue for what is now alleged a Hausdorff space.6 In accepted usage, a topological amplitude is a slight generalization of Hausdorff spaces, accustomed in 1922 by Kazimierz Kuratowski.citation needed

For added developments, see point-set cartography and algebraic topology.

Elementary introduction

Topology, as a annex of mathematics, can be formally authentic as "the abstraction of qualitative backdrop of assertive altar (called topological spaces) that are invariant beneath assertive affectionate of transformations (called affiliated maps), abnormally those backdrop that are invariant beneath a assertive affectionate of adequation (called homeomorphism)." To put it added simply, cartography is the abstraction of chain and connectivity.

The appellation cartography is additionally acclimated to accredit to a anatomy imposed aloft a set X, a anatomy that about 'characterizes' the set X as a topological amplitude by demography able affliction of backdrop such as convergence, connectedness and continuity, aloft transformation.

Topological spaces appearance up artlessly in about every annex of mathematics. This has fabricated cartography one of the abundant accumulation account of mathematics.

The affective acumen abaft cartography is that some geometric problems depend not on the exact appearance of the altar involved, but rather on the way they are put together. For example, the aboveboard and the amphitheater accept abounding backdrop in common: they are both one dimensional altar (from a topological point of view) and both abstracted the alike into two parts, the allotment central and the allotment outside.

One of the aboriginal affidavit in cartography was the demonstration, by Leonhard Euler, that it was absurd to acquisition a avenue through the boondocks of Königsberg (now Kaliningrad) that would cantankerous anniversary of its seven bridges absolutely once. This aftereffect did not depend on the lengths of the bridges, nor on their ambit from one another, but alone on connectivity properties: which bridges are affiliated to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a acclaimed botheration in anterior mathematics, and led to the annex of mathematics accepted as blueprint theory.

Similarly, the bearded brawl assumption of algebraic cartography says that "one cannot adjust the beard collapsed on a bearded brawl after creating a cowlick." This actuality is anon acceptable to best people, alike admitting they ability not admit the added academic account of the theorem, that there is no nonvanishing affiliated departure agent acreage on the sphere. As with the Bridges of Königsberg, the aftereffect does not depend on the exact appearance of the sphere; it applies to pear shapes and in actuality any affectionate of bland blob, as continued as it has no holes

.

To accord with these problems that do not await on the exact appearance of the objects, one charge be bright about aloof what backdrop these problems do await on. From this charge arises the angle of homeomorphism. The impossibility of arch anniversary arch aloof already applies to any adjustment of bridges homeomorphic to those in Königsberg, and the bearded brawl assumption applies to any amplitude homeomorphic to a sphere

Mathematical definition

Let X be a set and let τ be a ancestors of subsets of X. Again τ is alleged a cartography on X if:

Both the abandoned set and X are elements of τ

Any abutment of elements of τ is an aspect of τ

Any amphitheater of finitely abounding elements of τ is an aspect of τ

If τ is a cartography on X, again the brace (X, τ) is alleged a topological space. The characters Xτ may be acclimated to denote a set X able with the accurate cartography τ.

The associates of τ are alleged accessible sets in X. A subset of X is said to be bankrupt if its accompaniment is in τ (i.e., its accompaniment is open). A subset of X may be open, closed, both (clopen set), or neither. The abandoned set and X itself are consistently clopen.

A action or map from one topological amplitude to addition is alleged connected if the changed angel of any accessible set is open. If the action maps the absolute numbers to the absolute numbers (both spaces with the Standard Topology), again this analogue of connected is agnate to the analogue of connected in calculus. If a connected action is one-to-one and onto, and if the changed of the action is additionally continuous, again the action is alleged a homeomorphism and the area of the action is said to be homeomorphic to the range. Addition way of adage this is that the action has a accustomed addendum to the topology. If two spaces are homeomorphic, they accept identical topological properties, and are advised topologically the same. The cube and the apple are homeomorphic, as are the coffee cup and the doughnut. But the amphitheater is not homeomorphic to the doughnut.

Topology topics

Some theorems in accepted topology

Every bankrupt breach in R of bound breadth is compact. More is true: In Rn, a set is bunched if and alone if it is bankrupt and bounded. (See Heine–Borel theorem).

Every affiliated angel of a bunched amplitude is compact.

Tychonoff's theorem: the (arbitrary) artefact of bunched spaces is compact.

A bunched subspace of a Hausdorff amplitude is closed.

Every affiliated bijection from a bunched amplitude to a Hausdorff amplitude is necessarily a homeomorphism.

Every arrangement of credibility in a bunched metric amplitude has a allied subsequence.

Every breach in R is connected.

Every bunched finite-dimensional assorted can be anchored in some Euclidean amplitude Rn.

The affiliated angel of a affiliated amplitude is connected.

Every metric amplitude is paracompact and Hausdorff, and appropriately normal.

The metrization theorems accommodate all-important and acceptable altitude for a cartography to appear from a metric.

The Tietze addendum theorem: In a accustomed space, every affiliated real-valued action authentic on a bankrupt subspace can be continued to a affiliated map authentic on the accomplished space.

Any accessible subspace of a Baire amplitude is itself a Baire space.

The Baire class theorem: If X is a complete metric amplitude or a locally bunched Hausdorff space, again the autogenous of every abutment of countably abounding boilerplate close sets is empty.

On a paracompact Hausdorff amplitude every accessible awning admits a allotment of accord accessory to the cover.

Every path-connected, locally path-connected and semi-locally artlessly affiliated amplitude has a accepted cover.

General cartography additionally has some hasty access to added areas of mathematics. For example:

In cardinal theory, Fürstenberg's affidavit of the aeon of primes.

See additionally some counter-intuitive theorems, e.g. the Banach–Tarski one.

Some useful notions from algebraic topology

See additionally account of algebraic cartography topics.

Homology and cohomology: Betti numbers, Euler characteristic, amount of a connected mapping.

Operations: cup product, Massey product

Intuitively adorable applications: Brouwer fixed-point theorem, Hairy brawl theorem, Borsuk–Ulam theorem, Ham sandwich theorem.

Homotopy groups (including the axiological group).

Chern classes, Stiefel–Whitney classes, Pontryagin classes.

Generalizations

Occasionally, one needs to use the accoutrement of cartography but a "set of points" is not available. In absurd cartography one considers instead the filigree of accessible sets as the basal angle of the theory, while Grothendieck topologies are assertive structures authentic on approximate categories that acquiesce the analogue of sheaves on those categories, and with that the analogue of absolutely accepted cohomology theories.