Secondly, the topological space is a generalization of the geometry of the real number R, and it is an abstract space, in which there are not only numbers, but also other things, such as functions, matrices and so on.

In addition, the continuity of mapping is defined by the open set in topological space, while the definition of functional continuity in mathematical analysis is based on Euclidean norm. By defining Euclidean norm, we can define a series of concepts such as open set, closed set, convergence and continuity. Remember the ε -δ language? It just defines continuity by distance.

..... There is no norm in general topological space, only an open set. So the open set is defined as soon as the topology appears.

The definition of continuity in topological space is: x, y is topological space, f: x->; Y is a mapping, and the necessary and sufficient condition for F to be continuous in X is that for any open set U in Y, F inverse (U) is an open set in X. ..

Calculus on Manifold (Ouyang Guangzhong) p23 p89

Fundamentals and methods of topology (Noguchi Hiroshi).