Topology

Maurice Fréchet introduced the concept of metric space in 1906.

George Cantor, the inventor of set theory, studied extensively on limits.

In 1914, Hausdorff coined the term "topological space" and gave definition to what is now called Hausdorff space.

The current concept of topological space was described by Kuratowski in 1922.

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 connected together". One of the first papers in topology was the demonstration, by Leonhard Euler, 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. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. This problem, the Seven Bridges of Königsberg, is now a famous problem in introductory mathematics.

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair on a ball smooth". 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 exact shape of the sphere; it applies to pear shapes and in fact any kind of blob, as long as it has no holes.

In order 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 topological equivalence. The impossibility of crossing each bridge just once applies to any arrangement of bridges topologically equivalent to those in Königsberg, and the hairy ball theorem applies to any space topologically equivalent to a sphere. Formally, two spaces are topologically equivalent if there is a homeomorphism between them. In that case the spaces are said to be homeomorphic, and they are considered to be essentially the same for the purposes of topology.

Formally, a homeomorphism is defined as a continuous bijection with a continuous inverse, which is not terribly intuitive even to one who knows what the words in the definition mean. A more informal criterion gives a better visual sense: two spaces are topologically equivalent if one can be deformed into the other without cutting it apart or gluing pieces of it together. The traditional joke is that the topologist can't tell the coffee cup she is drinking out of from the donut she is eating, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

One simple introductory exercise is to classify the letters of the English alphabet according to topological equivalence. To be simple, it is assumed that the lines of the letters have nonzero width. Then in most fonts, there is a class {a,b,d,e,g,o,p,q} of letters with a hole, a class {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} of letters without a hole, and a class {i,j} of letters consisting of two pieces. For a more complicated exercise, it may be assumed that the lines have zero width; one can get several different classifications depending on which font is used.

Some useful notions from algebraic topology

See also list of algebraic topology topics.

• Homology and cohomology: Betti numbers, Euler characteristic.
• Nice applications: Brouwer Fixed Point Theorem, Borsuk-Ulam Theorem.
• Homotopy groups (including the fundamental group).
• Chern classes, Stiefel Whitney classes, Pontrjagin classes.

Sketchy outline of the deeper theory

• (Co)fibre sequences: Puppe sequence, computations
• Homotopy groups of spheres
• Obstruction theory
• K-theory: KO, algebraic K-theory
• Stable homotopy
• Brown representability
• (Co)bordism
• Signatures
• BP and Morava K-theory
• Surgery obstructions
• H-spaces, infinite loop spaces, A_∞ rings
• Homotopy theory of affine schemes
• Intersection cohomology

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.

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