What Is Topology?¶
Summary
Topology studies properties of geometric objects that are preserved under continuous deformation. Concepts such as homeomorphism, continuity, connectedness and compactness form the language in which Poincaré stated his conjecture in 1904.
1. Geometry without a ruler¶
Classical geometry measures lengths, angles and areas. Topology dispenses with every metric. It asks: Which properties of a space survive when one stretches, squeezes or bends it arbitrarily – just without tearing or gluing?
The standard anecdote: to a topologist, a coffee cup and a doughnut are the same object. Both surfaces have exactly one hole, and a continuous deformation carries one into the other. A sphere, by contrast, has no hole and is topologically distinct from cup and doughnut.
"Topology is the study of those properties of geometric objects that remain unchanged under continuous deformations." — John M. Lee, Introduction to Topological Manifolds (2011), p. 1
2. Continuity as a basic notion¶
In analysis, continuity is familiar as the \(\varepsilon\)-\(\delta\) property of functions \(\mathbb{R} \to \mathbb{R}\). Topologically it can be phrased more generally: a map \(f \colon X \to Y\) between two spaces is continuous if the preimage of every open set is open.
This definition only needs the notion of open set. A topological space is accordingly a set \(X\) together with a system of distinguished subsets – the open sets – satisfying three simple conditions: \(\emptyset\) and \(X\) are open, arbitrary unions of open sets are open, and finite intersections of open sets are open.
From this thin foundation all topological notions grow.
3. Homeomorphism – topological equality¶
When are two spaces topologically equal? When there is a homeomorphism between them: a bijection \(f \colon X \to Y\) such that \(f\) and \(f^{-1}\) are both continuous.
- A sphere and the boundary of a cube are homeomorphic: both are closed two-dimensional surfaces without a hole.
- An open interval \((0,1)\) is homeomorphic to \(\mathbb{R}\).
- A circle and a line segment are not homeomorphic: removing an interior point from the segment splits it into two pieces; removing a point from the circle leaves it connected.
The last example illustrates the typical proof technique: identify a property preserved under homeomorphism – a topological invariant – and use it to distinguish spaces.
4. Topological invariants¶
A topological invariant is a quantity or property that agrees on homeomorphic spaces. Three of the most basic ones:
Connectedness. A space is connected if it cannot be split into two disjoint non-empty open subsets. A circle is connected, a union of two disjoint circles is not.
Compactness. This generalises the idea "closed and bounded" to arbitrary topological spaces. A sphere is compact, a plane is not.
Dimension. Intuitively the number of independent directions. That it is indeed an invariant – \(\mathbb{R}^m\) is homeomorphic to \(\mathbb{R}^n\) if and only if \(m = n\) – is a non-trivial result (Brouwer 1911) and one of the founding achievements of algebraic topology.
Beyond these, algebraic invariants such as the fundamental group (see Article 03) and the homology and cohomology groups provide finer information. They assign to each space an algebraic structure that is preserved under homeomorphism.
5. A short history¶
Euler (1736). The Königsberg bridge problem and the polyhedron formula \(V - E + F = 2\) count as the first topological results – statements about combinatorial structure, independent of size or shape.
Riemann (1857). In his work on algebraic functions Riemann introduced the Riemann surfaces named after him and classified closed surfaces by their genus \(g\) – the number of handles.
Poincaré (1895–1904). In Analysis Situs Henri Poincaré laid the foundations of modern algebraic topology: fundamental group, homology, Betti numbers, triangulation. The paper ends with the question now known as the Poincaré Conjecture (see Article 04).
Hausdorff (1914). With Grundzüge der Mengenlehre Felix Hausdorff established the axiomatic definition of topological spaces in the form still used today.
"Henri Poincaré, more than any other person, is responsible for the emergence of topology as an independent branch of mathematics." — Allen Hatcher, Algebraic Topology (2002), preface
6. Why topology for Poincaré?¶
The Poincaré Conjecture is a purely topological statement: it says nothing about lengths, angles or curvature, only about the shape of a three-dimensional manifold (see Article 02). Concretely, it asserts that a closed 3-manifold in which every loop can be continuously contracted to a point must be homeomorphic to the 3-sphere \(S^3\).
Remarkable is the route the proof takes: Perelman and Hamilton translate the topological question via Riemannian metrics and the Ricci flow into differential geometry. The topological statement is obtained by controlling geometry – a pattern shared with Wiles's proof of Fermat's Last Theorem.
7. What comes next¶
The following articles introduce, systematically, the topological notions needed for the conjecture:
| Article | Topic |
|---|---|
| 02 – Manifolds | Locally Euclidean spaces, dimension, closed vs. with boundary |
| 03 – The Sphere and Simple Connectedness | \(S^n\), fundamental group, homotopy |
| 04 – What Is the Poincaré Conjecture? | Original 1904, generalisation, higher dimensions |
| 05 – Thurston's Geometrization Conjecture | Eight model geometries, Thurston |
Background knowledge
For a refresher on open sets, continuity or convergence, the Background Knowledge section provides entry points to set theory and analysis.
Sources¶
- Allen Hatcher: Algebraic Topology, Cambridge University Press (2002)
- John M. Lee: Introduction to Topological Manifolds, 2nd ed., Springer (2011)
- Henri Poincaré: Analysis Situs, Journal de l'École Polytechnique 1 (1895), 1–123
- Felix Hausdorff: Grundzüge der Mengenlehre, Veit & Comp., Leipzig (1914)