附录B 参考文献（原文）
See teaching point set topology
Topology is the study of geometry is invariant under continuous deformation properties. Topology of the concepts, theories and methods have been widely infiltrated into modern mathematics, science and social science in many fields, and has increasingly important applications, so learning the basics of topology, not only to provide the necessary modern mathematics learning basic knowledge, but from a higher point of view to observe, analyze the content of mathematics subjects, deepen awareness and understanding of the content. Because some of the basic concepts of topology is quite abstract for beginners, so it is necessary mathematical analysis of the linear space and some differences and relations between principle and thus play a multiplier effect.
First, the distinction between linear structure and isomorphic mapping, topology and explain the homeomorphism
Linear structure and topology of space of the two structures. In mathematics teaching and learning, to distinguish between the relationship and distinction between the two is not very easy for beginners thing, so in teaching, teachers should be based on the actual situation, clarifying the relationship between the it allows students to gain a deeper understanding of topological spaces and continuous maps related concepts.
In explaining the concept of topological space, we pointed out that topological space is a set of equipment, after the topology of space, topological space is the set of elements in the elements, topological properties of some open sets satisfy certain race, but if the equipment different topology has a different topological space. The linear space is a number of addition and multiplication satisfy the closed set.
For example: We often use the real number space R It can be seen as a linear space, its linear structure is defined in addition we usually several multiplication can also be viewed as a topological space, its topology is the real axis open set consisting of all open sets of the family, it satisfies the three topological nature, it is a special real topological linear space.
And if we define the set A = {1,2,3}, define topology = {{1,2,3}}, which is usually referred to mediocrity topology, topology open set is an empty set with itself, This is the smallest topology, if the definition of topology = {{1,2,3}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}} if it satisfies the three properties of topology, so (A,) is a topological space. If we define the open set family as
= {{1,2,3}, {1,2}, {1,3}, {2,3}}
It is not a topology, because of the intersection of two open sets open set does not belong to this family.
For two linear space is concerned, if there is a one mapping, can keep several multiplication addition, it is called an isomorphism. If there are two linear space isomorphism, the two spaces is isomorphic with the structure of the two spaces have the same linear structure. For finite dimensional linear space, with the structure of the two spaces have the same dimension, which can be seen as invariants of this space, it is pointed out that as long as the same dimensions, then their linear structure is the same.
As for the topological space is concerned, if there is a one mapping, and this mapping and its inverse mapping is continuous, is called homeomorphism. If there are two topological spaces homeomorphic map, then the two spaces are homeomorphic, homeomorphism of the two spaces have the same topology.
Therefore, understanding the difference between the two can help students better learn the theory of topological spaces. Topology is talking about under this homeomorphism of topological invariants, such as connectivity, countability, compactness and so on.
Second, the contact of mathematical analysis to explain the relevant principles of topology
Mathematical analysis is about the topology on the set of real numbers, so it had an immeasurable role in learning topology, we study and teach in the point set topology, it is necessary to contact the relevant conclusions of mathematical analysis to teach, which makes our Teaching content is not a hollow, and also to deepen the understanding of mathematical analysis, but also to learn a new theory and methods.
As we have said countability axiom, the real number may be an example of continuous function spaces, rational point if you know the analytical expression of the function, then we can based on real numbers and continuous functions dense to know the function of the analytical expression ; in spoken Hausdorff space, we know the number of columns or the uniqueness of the function extreme conclusion is obvious.
Another example is in teaching the content of compactness, you can first consider the real space in the closed interval, as the sub-range of real space, it has the following properties: every open cover has a finite sub-cover; each countable open covering is children have limited coverage; each sequence has a convergent subsequence; every infinite subset has a cluster point. Based on these properties, we can define four topological space: tight space, countably compact space, compact space sequence, sequentially compact space. These space from which we have had a strong desire for knowledge and interest in learning.
Third, with examples, focus on teaching methods
Point set topology is different from other undergraduate mathematics courses, such as mathematical analysis, advanced algebra, differential equations courses, almost no calculation like the content, logical, content abstraction; and more basic concept is, for is difficult for beginners, in many textbooks, the introduction of some concepts, followed by a lengthy series of theorems and proof of examples of small, some textbook examples also appear more abstract, if scripted, the stu
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