This article is about the books on topology pdf of mathematics. Problem and Polyhedron Formula are arguably the field’s first theorems. 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

By the middle of the 20th century, topology had become a major branch of mathematics. 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. Some authorities regard this analysis as the first theorem, signalling the birth of topology. German, in 1847, having used the word for ten years in correspondence before its first appearance in print. 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. Georg Cantor in the later part of the 19th century.

Topological spaces show up naturally in almost every branch of mathematics. This has made topology one of the great unifying ideas of mathematics. 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. 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 connect to which islands or riverbanks. 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.

Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. 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. The result depends partially on the font used. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent.