[Number of notes] on finite coverage theorem

1. Theorem: Set I is a bounded closed interval, {Uα} is an open overlay of I, then, S.T .

2, two key points:

(1) The coverage interval must be a closed interval

(2) The interval of covering closed interval and interval system must be open interval

3, the closed interval of this property, called the tightness

4, in the basic concept of topology, the most puzzling is the "tightness" (compactness), which describes a space or a set of "tight." The formal definition is "if any of the open overlays of a set have finite sub-overrides, then it is tight". At first glance, it's a bit confusing. What exactly does it want to describe? And the "tight" the adjective and how to relate it? An intuitive point of understanding, a few sets are "tight", that is, unlimited points scattered in, it is not possible to fully diffuse. No matter how small the neighborhood, there must be an infinite number of points in the neighborhood. The Mystery of this definition of compactness is in limited and infinite transformations. A tight collection is covered by an infinite number of small neighborhoods, but the finite one can always be found to cover the whole. So what are the consequences? An infinite number of points scattered in, there is always a neighborhood wrapped countless points. The neighborhood is so small-this guarantees that there are limits in the infinite sequence.

Although the concept of Compact is a bit less intuitive, it plays an extremely important role in the analysis. Because it relates to the existence of limits-this is the basis of mathematical analysis. The friends who understand functional analysis know that the sequence is convergent, and many times it is seen. In calculus, an important theorem-bounded sequences necessarily contain convergent sub-columns, which is the root of this.

- by a Daniel in MIT

5, the problem of tightness, can be said to be a very important problem in the topology. For real numbers, several lemma related to the tightness of the closed interval, such as limited coverage, closed interval sets, and Cantor Limit-point lemma. There are some generalizations about the general topological space. These generalizations are a good portrayal of firmness. The understanding of the lemma itself is very good: it is not enough to Sakai the dots. It is because "cannot be fully Sakai", so there must be limited open coverage. Compact is a topological property, the analysis of the topological properties of the analysis of the nature of the examples are too many, and many are related to this compact. For example, the most basic, continuous function on the compact set can achieve maximum and minimum value. It is clear that the issue of non-compact collections is not necessarily true.

6, some of my understanding:

(1) since it is "arbitrary - must" Relationship, then we don't have to consider the existence of an outside point. Because for example [a,b] need to be covered, Just cover it with a large open interval , but there's no need to study it. We consider those situations that are not so "obvious". is within the point of the field, it can be known that is certainly can be completely covered [a,b] [a,b]

(2) has an intuitive metaphor. Imagine the real number of adults, standing in [a,b] This is raining, Everyone's umbrella is sure to keep everyone from getting wet. But we can choose a limited personal umbrella, the rest of the people do not umbrella, the same can ensure that everyone will not be wet. And if This is raining, a end this person does not need an umbrella, but from the right a near the process there are infinite individuals, I choose someone the umbrella, on the left there is always someone will rain, but just cover off

(3) Give an example to explain the above the situation. For example, consider the interval we use to overwrite it, and, when known, to cover the and up for , because it can be covered to

(4) The closed interval is not covered by the closed interval. First, the open cover theorem corresponds to the tightness in functional analysis. The key is that each point in the opening interval is an inner point (here I want to say, if an opening interval is to overwrite a point x , then it must not just cover the x, but the x field), and the closed interval is not necessarily (the closed interval here includes a single point set). Then a [ A, b] can be fully covered by all the single points on [a, b], but this is infinite, and arbitrary culling of one will result in [A, b] cannot be completely overwritten.

[Number of notes] on finite coverage theorem