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A three-dimensional model of a figure-eight knot. The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 41.

Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

The ideas underlying topology go back to Gottfried Wilhelm Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. 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.

Motivation

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Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one-dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside.

In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg (now Kaliningrad) that would cross each of its seven bridges exactly once.[1] This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory.[2]

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick."[3] This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes.

To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.

A continuous transformation can turn a coffee mug into a donut.
Ceramic model by Keenan Crane and Henry Segerman.

Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A famous example, known as the "Topologist's Breakfast", is that a topologist cannot distinguish a coffee mug from a doughnut.[4] A pliable torus (shaped like a doughnut) can be reshaped to a coffee mug by creating a dimple and progressively enlarging it while shrinking the central hole into the mug's handle.[5]

Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.

History

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The Seven Bridges of Königsberg was a problem solved by Euler.
See also: History of the separation axioms

Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.[6] Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology.[6] On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron. This led to his polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology.[7]

Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.[8] Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print.[9] The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated".[10]

Their work was corrected, consolidated and greatly extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology.[8]

Topological characteristics of closed 2-manifolds[8]
Manifold Euler number Orientability Betti numbers Torsion coefficient
(1-dim)
b0 b1 b2
Sphere 2 Orientable 1 0 1 none
Torus 0 Orientable 1 2 1 none
2-holed torus −2 Orientable 1 4 1 none
g-holed torus (genus g) 2 − 2g Orientable 1 2g 1 none
Projective plane 1 Non-orientable 1 0 0 2
Klein bottle 0 Non-orientable 1 1 0 2
Sphere with c cross-caps (c > 0) 2 − c Non-orientable 1 c − 1 0 2
2-Manifold with g holes
and c cross-caps (c > 0)		 2 − (2g + c) Non-orientable 1 (2g + c) − 1 0 2

The development of topology in the 20th century was marked by significant advances in both foundational theory and its application to other fields of mathematics. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906.[11] A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and defined what is now called a Hausdorff space.[12] Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.[13]

Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series. For further developments, see point-set topology and algebraic topology.

The 2022 Abel Prize was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects".[14]

Concepts

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Topologies on sets

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Main article: Topological space

The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be thought of as the same set with different topologies.

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:

  1. Both the empty set and X are elements of τ.
  2. Any union of elements of τ is an element of τ.
  3. Any intersection of finitely many elements of τ is an element of τ.

If τ is a topology on X, then the pair (X, τ) is called a topological space. The notation Xτ may be used to denote a set X endowed with the particular topology τ. By definition, every topology is a π-system.

The members of τ are called open sets in X. A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x.

Continuous functions and homeomorphisms

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A continuous deformation (a type of homeomorphism) of a mug into a doughnut (torus) and of a cow into a sphere
Main articles: Continuous function and homeomorphism

A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If a continuous function is one-to-one and onto, and if the inverse of the function is also continuous, then the function is called a homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to the doughnut.

Manifolds

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Main article: Manifold

While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds. Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane, which cannot (that is, all their realizations in three dimensions are surfaces that are not manifolds).

Topics

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General topology

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Main article: General topology

General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology.[15][16] It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby, arbitrarily small, and far apart can all be made precise by using open sets. Several topologies can be defined on a given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected.

Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line, the complex plane, real and complex vector spaces and Euclidean spaces. Having a metric simplifies many proofs.

Algebraic topology

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Main article: Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces.[17] The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

The most important of these invariants are homotopy groups, homology, and cohomology.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

Differential topology

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Main article: Differential topology

Differential topology is the field dealing with differentiable functions on differentiable manifolds.[18] It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

More specifically, differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Geometric topology

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Main article: Geometric topology

Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology.[19] Some examples of topics in geometric topology are orientability, handle decompositions, local flatness, crumpling and the planar and higher-dimensional Schönflies theorem.

In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory.

Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries.

2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.

Generalizations

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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,[20] while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those categories and with that the definition of general cohomology theories.[21]

Applications

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Biology

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Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on the pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory, a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis.[22]

Computer science

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Topological data analysis uses techniques from algebraic topology to determine the large-scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is to:

  1. Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
  2. Analyse these topological complexes via algebraic topology – specifically, via the theory of persistent homology.[23]
  3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number, which is called a barcode.[23]

Several branches of programming language semantics, such as domain theory, are formalized using topology. In this context, Steve Vickers, building on work by Samson Abramsky and Michael B. Smyth, characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties.[24]

Physics

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Topology is relevant to physics in areas such as condensed matter physics,[25] quantum field theory, quantum computing and physical cosmology.

The topological dependence of mechanical properties in solids is of interest in the disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials.[26] The compressive strength of crumpled topologies is studied in attempts to understand the high strength to weight of such structures that are mostly empty space.[27] Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics.

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory, the theory of four-manifolds in algebraic topology, and the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

The topological classification of Calabi–Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.[28]

In topological quantum computers, the qubits are stored in topological properties, that are by definition invariant with respect to homotopies.[29]

In cosmology, topology can be used to describe the overall shape of the universe.[30] This area of research is commonly known as spacetime topology.

In condensed matter, a relevant application to topological physics comes from the possibility of obtaining a one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect, and then generalized in other areas of physics, for instance in photonics.[31] David Thouless, Duncan Haldane, and Michael Kosterlitz were awarded the 2016 Nobel Prize in Physics for their work on Topological orders.[32]

Robotics

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The possible positions of a robot can be described by a manifold called configuration space.[33] In the area of motion planning, one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose.[34]

Games and puzzles

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Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components.[35][36][37]

Fiber art

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In order to create a continuous join of pieces in a modular construction, it is necessary to create an unbroken path in an order that surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path.[38]

Resources and research

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Major journals

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Major books

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  • Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-181629-9
  • Willard, Stephen (2016). General topology. Dover books on mathematics. Mineola, N.Y: Dover publications. ISBN 978-0-486-43479-7
  • Armstrong, M. A. (1983). Basic topology. Undergraduate texts in mathematics. New York: Springer-Verlag. ISBN 978-0-387-90839-7
  • John Kelley "General Topology" Springer, 1979.

See also

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References

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Further reading

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Topology

View on Grokipedia
from Grokipedia
Topology is a branch of concerned with abstract notions of continuity that generalize the notion of to the more general setting of . Often described as "rubber-sheet geometry," it generalizes classical by abstracting away rigid notions of distance and angle, focusing instead on qualitative features like connectivity, continuity, and the ability to distinguish shapes topologically. This field provides a foundational framework for understanding spatial structures in a flexible manner, underpinning much of modern including , , and physics. The development of topology traces its roots to the 18th century, with early insights emerging from problems in and polyhedral . In 1736, Leonhard Euler analyzed the Seven Bridges of Königsberg problem, effectively introducing concepts of connectivity that later became central to . By 1752, Euler established the formula V−E+F=2V - E + F = 2 for convex polyhedra, where VV is the number of vertices, EE edges, and FF faces, laying groundwork for the as a topological invariant. The saw further advancements, including Augustin-Louis Cauchy's 1821 definition of continuity and his 1825 work on complex integrals introducing concepts akin to winding numbers, and Bernhard Riemann's 1857 introduction of Riemann surfaces to handle multi-valued functions, which highlighted as a key . Topology coalesced as a distinct discipline in the late 19th and early 20th centuries, with Henri Poincaré's 1904 formulation of the on the classification of three-dimensional manifolds, which was proved in 2003 by , marking a pivotal moment in its evolution from geometric intuition to rigorous abstraction. At its core, topology is formalized through the concept of a , defined as a set XX equipped with a collection T\mathcal{T} of subsets called , satisfying three axioms: the and XX are open; arbitrary unions of open sets are open; and finite intersections of open sets are open. These open sets capture the notion of "nearness" without metrics, enabling the study of closed sets (complements of open sets), neighborhoods, and limit points. A key notion is that of continuous functions between topological spaces: a f:X→Yf: X \to Y is continuous if the preimage of every open set in YY is open in XX, preserving structural properties across spaces. Two spaces are homeomorphic if there exists a continuous with a continuous inverse, meaning they are topologically indistinguishable, as in the case of a and an . Topology branches into several subfields, each emphasizing different tools and applications. (or point-set topology) deals with foundational concepts like , connectedness, and separation axioms, providing the analytical groundwork for spaces beyond the Euclidean. employs algebraic invariants, such as fundamental groups and homology groups, to classify spaces and detect holes or connectivity, with roots in Poincaré's work. integrates smooth structures on manifolds, studying embeddings and immersions, while focuses on low-dimensional manifolds and . These areas intersect with fields like , dynamical systems, and quantum physics, where topological invariants model phenomena from particle interactions to .

Motivation and Overview

Defining Topology

Topology is the branch of that studies topological spaces, abstract structures consisting of a set equipped with a collection of subsets called open sets, which satisfy three axioms: the and the entire space are open, arbitrary unions of open sets are open, and finite intersections of open sets are open. These open sets provide the foundation for defining neighborhoods around points, enabling the precise characterization of concepts like convergence, limits, and continuity in a manner independent of any underlying metric or . A key abstraction in topology is its departure from classical geometry by disregarding quantitative measures such as distances, angles, and straight lines, instead emphasizing qualitative properties that remain invariant under continuous deformations—like stretching, twisting, or bending, but not tearing or gluing. This focus on intrinsic notions of nearness and continuity allows topology to capture the essential "shape" of spaces in the broadest sense, generalizing geometric ideas to non-Euclidean and even non-geometric settings. A representative example of topological equivalence is the homeomorphism between a and a (torus), where both objects can be continuously deformed into one another because they possess the same topological features, such as a single hole threading through the structure, illustrating how topology preserves such invariants while ignoring superficial differences in form. The term "topology" was coined by the German mathematician in his 1847 work Vorstudien zur Topologie, derived from the Greek words topos (place or location) and logos (study or discourse), reflecting its concern with the arrangement and positioning of mathematical objects.

Importance in Mathematics and Beyond

Topology serves as a foundational framework that unifies diverse branches of mathematics, including , , and , by providing essential tools for studying limits, continuity, and topological invariants. In , topological concepts underpin the rigorous definition of continuity and convergence in metric spaces, enabling the generalization of to abstract settings. Geometry benefits from topology's emphasis on intrinsic properties preserved under deformations, bridging Euclidean and non-Euclidean structures through notions like manifolds. Algebra intersects via homological methods, where topological spaces are assigned algebraic invariants such as homology groups, facilitating the classification of shapes and the study of symmetries. This unifying power is exemplified by topology's "rubber-sheet " perspective, which models shapes that can deform continuously without tearing or gluing, capturing essential qualitative features invariant under such transformations. This approach appeals across disciplines by allowing mathematicians to abstract spatial problems, focusing on connectivity and holes rather than rigid metrics, thus enabling solutions to problems in higher dimensions or irregular forms that defy traditional geometric tools. A landmark illustration of topology's depth is the resolution of the , proven by in 2003 using techniques, which established that every simply connected, closed is homeomorphic to the , profoundly advancing the classification and understanding of three-dimensional manifold structures. In the , topology's relevance has extended to (TDA), a field that applies to extract robust features from noisy, high-dimensional datasets, revealing underlying shapes like clusters and loops that persist across scales and aiding in applications from to scientific computing.

Historical Development

Early Foundations

The origins of topology can be traced to 18th-century problems in geometry that emphasized connectivity and qualitative properties over precise measurements. A seminal precursor was Leonhard Euler's 1736 solution to the Seven Bridges of Königsberg problem, which sought a path crossing each of the city's seven bridges exactly once and returning to the starting point. Euler demonstrated that no such path existed by modeling the landmasses as vertices and bridges as edges, introducing the idea that traversability depends on the degrees of vertices—an insight that laid foundational groundwork for and topological notions of connectivity. In the late 18th and early 19th centuries, mathematicians began exploring curved surfaces and higher-dimensional structures, bridging and . , in the 1790s, showed early interest in topological ideas through his study of and surfaces, compiling lists of knot diagrams as early as 1794, though he did not publish these works; his unpublished notes influenced later developments in surface classification. Building on such ideas, in the 1850s introduced the concept of manifolds in his 1854 habilitation lecture, describing them as multi-dimensional analogs of surfaces that generalize while preserving local properties like differentiability. Riemann's manifolds provided a framework for understanding connectivity in complex domains, such as Riemann surfaces for algebraic functions. The late 19th century saw the emergence of qualitative geometry, distinct from metric-based approaches. Henri Poincaré's series of papers in the 1880s, culminating in his 1895 "Analysis Situs," established the field as a study of spatial forms invariant under continuous deformations, introducing the as a way to capture loops in a space that cannot be continuously shrunk to a point—thus quantifying connectivity without coordinates. This work formalized "analysis situs" as a branch focused on intrinsic properties like holes and linking. Concurrently, coined the term "topology" in his 1847 book Vorstudien zur Topologie, using it to describe the study of position and neighborhood relations on surfaces, including early descriptions of non-orientable surfaces like the Möbius band in 1861. Complementing these geometric advances, in the 1870s developed set-theoretic tools for continuity, defining limit points and derived sets in 1872 to analyze point collections on the real line, which provided the abstract foundations for modern point-set topology.

20th Century Advancements

The early marked a pivotal shift toward axiomatic rigor in topology, with Felix Hausdorff's 1914 monograph Grundzüge der Mengenlehre providing the first systematic definition of topological spaces through neighborhood systems. Hausdorff's framework specified axioms for neighborhoods that ensured properties like symmetry and transitivity, allowing for a precise characterization of continuity without relying on metric structures. This innovation directly incorporated what is now known as the Hausdorff , requiring that any two distinct points possess disjoint neighborhoods, which became a cornerstone for subsequent separation axioms (T0 through T4) that classify topological spaces by their ability to distinguish points. A landmark result in this era was Luitzen Egbertus Jan Brouwer's , proved in 1911, which states that any from a closed n-dimensional to itself has at least one fixed point. This theorem, established through degree theory in , had profound implications, including Brouwer's 1912 theorem, which asserts that a continuous injective from an open subset of Euclidean n-space to itself is an open map, thereby preserving dimension under homeomorphisms. These results solidified the invariance of topological dimension and influenced fields beyond , such as fixed-point theory in . Institutional developments further propelled topology's growth, particularly through the topology group formed in the 1930s under Solomon Lefschetz's leadership. Lefschetz, who joined Princeton in 1924 and became a dominant figure, popularized the term "topology" in his 1930 book Topology, which synthesized combinatorial and algebraic approaches, and advanced via and fixed-point theorems like the Lefschetz formula. His work fostered a vibrant research community at Princeton, attracting figures like Norman Steenrod and James Hurewicz, and emphasized algebraic tools for studying manifolds, bridging and topology. Mid-century advancements in were epitomized by Stephen Smale's theorem, first proved in 1962, which demonstrates that a simply connected between manifolds of at least 5 is to a product . This breakthrough, relying on handle decompositions and , resolved the in high dimensions and enabled the classification of smooth manifolds up to , transforming the study of exotic structures in .

Core Concepts

Topological Spaces and Bases

A consists of a set XX and a collection τ\tau of subsets of XX, with the elements of τ\tau called open sets, such that τ\tau satisfies three axioms: the ∅\emptyset and XX itself belong to τ\tau; the union of any arbitrary collection of sets in τ\tau belongs to τ\tau; and the of any finite collection of sets in τ\tau belongs to τ\tau. This axiomatic framework, equivalent to systems based on neighborhood filters or closure operators, provides the foundational structure for abstracting notions of continuity and proximity without relying on metrics. The modern open-set formulation aligns closely with the neighborhood axioms introduced by , who emphasized systems where each point has neighborhoods satisfying closure under finite intersections and containing smaller neighborhoods, ensuring compatibility with limit point concepts. An alternative characterization uses Kuratowski's closure axioms, where a closure operator cl:P(X)→P(X)\mathrm{cl}: \mathcal{P}(X) \to \mathcal{P}(X) (with P(X)\mathcal{P}(X) the power set of XX) satisfies: cl(∅)=∅\mathrm{cl}(\emptyset) = \emptyset; A⊆cl(A)A \subseteq \mathrm{cl}(A) for all A⊆XA \subseteq X; cl(cl(A))=cl(A)\mathrm{cl}(\mathrm{cl}(A)) = \mathrm{cl}(A); and cl(A∪B)=cl(A)∪cl(B)\mathrm{cl}(A \cup B) = \mathrm{cl}(A) \cup \mathrm{cl}(B). These axioms, proposed by Kazimierz Kuratowski in 1922, generate a unique topology where the closed sets are the fixed points of the closure operator, offering a dual perspective to the open-set definition. Bases and subbases serve as generators for topologies, simplifying the description of open sets. A base B\mathcal{B} for a topology τ\tau on XX is a subcollection of τ\tau such that every set in τ\tau is a union of elements from B\mathcal{B}, and B\mathcal{B} satisfies two conditions: ⋃B∈BB=X\bigcup_{B \in \mathcal{B}} B = X; and for any B1,B2∈BB_1, B_2 \in \mathcal{B} with x∈B1∩B2x \in B_1 \cap B_2, there exists B3∈BB_3 \in \mathcal{B} such that x∈B3⊆B1∩B2x \in B_3 \subseteq B_1 \cap B_2. This structure allows topologies to be specified via a "small" collection of basic open sets, with the full topology obtained by taking all possible unions. A subbase S\mathcal{S} is a collection whose finite intersections form a base for τ\tau, providing an even coarser generator often useful for product or topologies. Illustrative examples highlight these concepts. The discrete topology on XX has τ=P(X)\tau = \mathcal{P}(X), where every subset is open; a base consists of all singletons {x}\{x\} for x∈Xx \in X, as any subset is their union. In contrast, the indiscrete (or trivial) topology has τ={∅,X}\tau = \{\emptyset, X\}, with {X}\{X\} serving as a base since the only non-empty open set is XX itself. The on the real numbers R\mathbb{R}, induced by the standard ordering, uses open intervals (a,b)(a, b) as a base, generating the familiar where open sets are unions of such intervals. These foundational elements establish the abstract framework for topology, enabling the study of properties invariant under continuous mappings, such as connectedness and compactness, without delving into specific metrics or algebraic structures.

Continuity and Homeomorphisms

In topology, continuity is defined without reference to distances or metrics, generalizing the epsilon-delta notion from analysis to arbitrary topological spaces. Specifically, given topological spaces (X,TX)(X, \mathcal{T}_X) and (Y,TY)(Y, \mathcal{T}_Y), a function f:X→Yf: X \to Y is continuous if, for every open set V∈TYV \in \mathcal{T}_Y, the preimage f−1(V)f^{-1}(V) is an open set in TX\mathcal{T}_X. This definition captures the intuitive idea that continuous functions preserve the structure of open neighborhoods, ensuring that points close in the domain map to points that remain "close" in a topological sense, without quantifying closeness. This formulation, introduced by Felix Hausdorff, allows continuity to be studied in diverse spaces beyond Euclidean ones, such as discrete or indiscrete topologies. Homeomorphisms extend this by providing a notion of topological equivalence between spaces. A homeomorphism is a bijective f:X→Yf: X \to Y whose inverse f−1:Y→Xf^{-1}: Y \to X is also continuous, meaning both ff and f−1f^{-1} map open sets to open sets. The term was coined by in his foundational work on analysis situs, emphasizing mappings that preserve all topological properties without distortion. Being homeomorphic is an on the class of topological spaces: it is reflexive (via the identity map), symmetric (since (f−1)−1=f(f^{-1})^{-1} = f), and transitive (composing two homeomorphisms yields another). Thus, homeomorphisms partition topological spaces into equivalence classes, where spaces within the same class are indistinguishable topologically, such as a and an , which can be continuously deformed into each other. Illustrative examples highlight these concepts. The identity map idX:X→X\mathrm{id}_X: X \to X, defined by idX(x)=x\mathrm{id}_X(x) = x for all x∈Xx \in X, is a trivial , as it is bijective and both it and its inverse (itself) are continuous in any topology. In contrast, consider the i:R2∖ℓ↪R2i: \mathbb{R}^2 \setminus \ell \hookrightarrow \mathbb{R}^2, where ℓ\ell is the xx-axis; this is continuous but not a , as it fails to be bijective (not surjective), and moreover, R2∖ℓ\mathbb{R}^2 \setminus \ell and R2\mathbb{R}^2 are not homeomorphic because the former is disconnected (union of two open half-planes) while the latter is connected. Such examples underscore that homeomorphisms preserve intrinsic topological features like connectedness. A key result linking continuity and homeomorphisms is Brouwer's invariance of domain theorem, which asserts that if U⊆RnU \subseteq \mathbb{R}^n is open and f:U→Rnf: U \to \mathbb{R}^n is a continuous injective map (hence a onto its image), then f(U)f(U) is open in Rn\mathbb{R}^n. Published by Luitzen Egbertus Jan Brouwer in 1912, this theorem demonstrates that homeomorphisms in Euclidean spaces preserve openness, with profound implications for distinguishing non-homeomorphic subsets, such as confirming that no exists between Rn\mathbb{R}^n and Rm\mathbb{R}^m for n≠mn \neq m.

Basic Properties: Connectedness and Compactness

In topology, a space XX is defined to be connected if it cannot be expressed as the union of two disjoint nonempty open sets. This property captures the intuitive notion of a space being "in one piece," preventing separation into independent components by the topology. Equivalently, the only subsets of XX that are both open and closed (clopen) are the and XX itself. A stronger condition is path-connectedness: a space XX is path-connected if, for any two points x,y∈Xx, y \in X, there exists a γ:[0,1]→X\gamma: [0,1] \to X such that γ(0)=x\gamma(0) = x and γ(1)=y\gamma(1) = y. Every path-connected space is connected, but the converse does not hold, as demonstrated by examples like the . Classic examples illustrate these concepts. The real line R\mathbb{R} with the standard topology is connected, as any nonempty open sets in R\mathbb{R} must overlap or one must contain an interval that connects points across the space. In contrast, the rational numbers Q\mathbb{Q} as a subspace of R\mathbb{R} are disconnected; for instance, they can be partitioned into Q∩(−∞,2)\mathbb{Q} \cap (-\infty, \sqrt{2})
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