Topology is an important branch of geometry developed in the19th century. As early as Euler or earlier, topology had sprouted. The famous "Seven Bridges over the cornices" and "Preliminary Topology of Mobius Ding". Christine was a student of Gauss, and after 1834, she was a professor at Kogan University. He wants to call this subject "positional geometry", but the name Todd is used to refer to projective geometry. So I used the name "topology" instead. The literal translation of "topology" means geography, which is similar to topography and geomorphology. In 1956, it is defined as topology in unified mathematical terms.

Although topology is a branch of geometry, this geometry is different from the usual "plane geometry" and "solid geometry". Usually the research object of plane geometry or solid geometry is the position of organs between points, lines and surfaces and the nature of examination questions. The content of topology research has nothing to do with the length, ocean, area and volume of the research object and the relationship between the nature and quantity of the test questions.

For example, in the usual plane geometry, if one figure on the plane is moved to another figure, if the two figures are completely coincident, then the two figures are called congruent figures, that is to say, the usual plane geometry is a discipline that studies the size and shape of the moving figure, while the figure studied in topology changes in the movement, regardless of its size and shape. In topology, there are no inflexible elements, and the size and shape of each figure are changing.

After Christine, Riemann introduced the concept of topology into the theory of complex variable functions and developed it into Riemann surface theory.

Early topology is clearly divided into two branches: one is point set topology, starting with Cantor's contribution; The other is combinatorial topology, which was initiated by Poincare at the end of last century. Poincare is usually slow, clumsy and has poor eyesight, which often gives people the impression of being absent-minded However, Poincare has extraordinary mental arithmetic and mathematical thinking ability. Pang Guolai had a great influence on mathematics in the 20th century. 1895 published analysissitus, which systematically discussed the contents of topology for the first time. Later, it developed into a fruitful branch of topology in the 20th century, and Poincare's research field was very extensive. His lectures at the University of Paris include capillary science, elasticity, thermodynamics, optics, electricity, cosmology and so on. In mathematics, he also involves non-Euclidean geometry, invariant theory and analytical mechanics, including probability theory.

Topology is a new subject As soon as it appeared, it quickly penetrated into all fields.