When it comes to topology, the rise of functional analysis is indispensable, especially the establishment of Hilbert space and Bannach space, which promotes the research of point set as space. The core problem of mathematical analysis is limit, and convergence and continuity are the basic problems of limit. In order to extend the study of convergence and continuity to general sets, it is necessary to describe the concept of "proximity" of points or sets on general sets. "Far" can be used to describe "near", but "far" and "near" are not necessarily related.

In fact, we also need to use topology to define "neighborhood" in advanced mathematics. For the nonempty set X, it is stipulated that every point of X has a subset family, which consists of subsets containing the point, and this subset family satisfies a set of neighborhood axioms (that is, a set of attributes given by imitating the characteristics of Euclidean space).

Each set in a subset family is called the neighborhood of a point. This gives a topology of X, and the X connected with this topology is called topological space. Every point of x has a neighborhood, so we can study the neighborhood of the point. Therefore, the concept of continuous mapping between two topological spaces can be defined by imitating calculus. If a mapping is continuous and there is an inverse mapping, and the inverse mapping is also continuous, it is called homeomorphism mapping. Two topological spaces with homeomorphism mapping are called homeomorphisms (intuitively speaking, the graphs corresponding to these two spaces change from one continuous terrain to another).

In order to prove the homeomorphism of two spaces, we only need to find the homeomorphism mapping between them. On the Euclidean line, as a subspace, two arbitrary closed intervals are homeomorphic; Any two open interval homeomorphisms; The semi-open and semi-closed interval [C, D] is homeomorphic to [a, b]. Point s2-p is excavated from a two-dimensional sphere and is homeomorphic to Euclidean plane K2. In order to prove that two topological spaces are different, it is necessary to prove that there is no homeomorphism mapping between them.

This method is to find homeomorphism invariants or topological invariants (that is, properties that remain unchanged under homeomorphism mapping); If the first space has homeomorphism invariants and the other space does not, then the two spaces are different. The common topological invariants in general topology are connectivity, road connectivity, compactness, column compactness, separation and so on. (See Topological Space).

Therefore, topology actually contains the contents of set theory, graph theory and functional analysis, and it is also a comprehensive subject, so it will be easier to lay a good foundation.