Topology is the study of those properties of a space that survive continuous deformation, and its guiding principle is that two spaces which can be continuously deformed into one another are equivalent for every question the subject asks. A homeomorphism is a bijection that is continuous in both directions, and by the definition of the subject the question whether two spaces are homeomorphic is the fundamental one. The subject was founded in practice by Euler, who classified curves and surfaces by counting features rather than by measuring, and was made rigorous by Poincare in 1895 in the Analysis Situs, where he introduced the fundamental group and homology as invariants of a space. The groundwork beneath it is point set topology, which asks what continuity, compactness, connectedness, and dimension mean in a general space, and which pays for itself because almost every later proof needs a compactness or separation argument. Compactness admits the finite subcover formulation, connectedness is captured by the absence of a separation into two open sets, and the separation axioms lead to a useful ladder of spaces from a plain set through a metric space to a compact metric space, which is where most of the interesting examples live. The classification of compact surfaces of genus zero and above then becomes the first great theorem of the subject, derived by cutting a surface into pairs of pants and reading the answer off the resulting equation relating the numbers of handles, boundary components, and punctures. The first of the two great invariants is the fundamental group, written pi 1, which records the ways loops in a space can be continuously deformed into one another. One obtains it by fixing a base point, considering all loops there, and declaring two of them equivalent when one can be deformed into the other through loops that stay based, which is an equivalence relation by a short concatenation argument. The group law is concatenation, and the identity is the constant loop. The fundamental group of a circle is the group of the integers, since a loop is classified by how many times it winds around, while the fundamental group of a sphere is trivial because every loop can be shrunk. The van Kampen theorem provides the fundamental computational tool, showing that the fundamental group of a space assembled from two pieces along a connected overlap can be built from the fundamental groups of the pieces by a quotient, which is what makes it possible to compute the group of a complicated space inductively. Covering spaces are the second tool: a space may be studied through a cover in which loops become transparent, and for a circle the cover given by the infinite line makes the winding number visible as a displacement. Higher homotopy groups appear when one asks about spheres of every dimension mapping into the space, and Whitehead showed that a map between simply connected complexes which induces isomorphisms on all these groups is itself a homotopy equivalence, so the homotopy groups genuinely determine the shape of a space. The second invariant is homology, which is built from a chain complex and is far easier to compute. A simplicial complex is assembled from vertices, edges, triangles, and higher simplices, and one forms the free abelian group generated by oriented simplices of each dimension, called the chain group. The boundary of a triangle is the sum of its edges with appropriate signs, and this defines a map from chains of dimension one higher to chains of the next lower dimension. The kernel of that map is the group of cycles, which are formal sums of simplices that bound nothing, and the image is the group of boundaries, which are chains that do bound. The homology group in dimension a is the first quotient by the second, and since every boundary is a cycle the quotient is well defined. The remarkable fact is that this quotient is invariant under subdivision of the complex and under large classes of deformation, so it is a property of the space rather than of the description, and it can be computed by linear algebra over a field, which is why the subject is now algorithmic. The numerical shadows of the homology groups are the Betti numbers, which count the rank of the group in each dimension, and their alternating sum is the Euler characteristic. The group in dimension zero counts connected components, the group in dimension one counts independent cycles, and the group in dimension two counts independent two dimensional holes, and the pattern continues. For a sphere the homology is the integers in dimensions zero and n with everything else trivial, while for a torus there are two independent cycles in dimension one and one in dimension two, and for a wedge of circles the pattern is exactly as many generators as there are circles, each of which appears as a reduced graph in the theory of graphs. This gives an immediate computational tool in engineering: the first Betti number of a graph equals its number of edges minus its number of vertices plus its number of connected components, so the cyclic structure of a network is a single subtraction away. Cohomology, defined as the dual of homology or equivalently by cochains and coboundaries, supports a product called the cup product which encodes how holes sit inside one another, and the universal coefficient theorem relates cohomology to homology over any field. Mayer and Vietoris give an exact sequence for a union of two subspaces, duality theorems relate a manifold to its complement, and the Euler characteristic of a closed manifold of dimension n equals the alternating sum of its Betti numbers, which for an oriented surface is the number two minus twice the genus. The reach of these ideas into applications is broad and increasing. In robotics the configuration space of a machine is a manifold or a stratified space, and the question whether a motion exists between two poses is a question about path connectedness, which in two dimensions means deciding whether the first Betti number is positive. Computer vision uses Morse theory and Reeb graphs, which are quotients of a space by a height function and summarise the shape of an object by a graph. Physics uses homology to organise the configuration spaces of scattering, where topology decides which states exist. Learning the subject gives a student a working notion of when two objects differ in a way that no measurement can detect, which is a habit worth having in every later branch.
