How to prove that the measurement space of compaction must be divisible

1.First of all, the compact measurement space must be the lindeloff space. A lindeloff space is a complete metric space with enough open coverages that a countable open coverage can be found.

2.Then, due to the countable nature of the open cover of the Lindeloff space, a countable dense subset can be found.

3.The arbitrary opening coverage of the compact metric space can be covered by a finite number of tees with a radius of 1 n, and a countable sub-overlay can be found by taking the family of tee sets with a radius of 1 n centered on all points as the full-space open coverage.

4.It follows that a compact metric space must be embedded in a Hilbert space. Since the hilbert space is divisible, the compact measurement space must also be divisible.

Therefore, we can conclude that the compact measurement space must be divisible.

First: a tight subset in a metric space is equivalent to a bounded closed set.

secondly: the union of finite bounded sets is a bounded set; And the union of a finite number of closed subsets is a closed set.

Therefore, a finite compact subset is a bounded closed set, that is, a compact set.

This is one of the basic propositions of metric space theory, which can be proved by the finite coverage of compact sets or the closed properties of columns.

In the metric space, the concept of convergence of the column of points can be defined by distance: xn x0 is d(xn,x0). The point column is called the Cauchy point column, which means that for any positive real number, there are natural numbers n, so that m, n n can prove that the convergence point column must be a Cauchy point column, and the reverse is not true.

The metric space in which each Cauchy point column converges is called a complete metric space. There are many good properties in this type of space. For example, the principle of compression mapping in a complete metric space holds.

It can be used to prove a series of existential uniqueness theorems for differential equations, integral equations, and systems of infinite linear algebraic equations. Any subset of the metric space x y with the original distance also becomes the metric space, called the subspace of x. If each tee shot {x x|d(x0,x)<>

Completeness definition: Any Cauchylie has a convergence point, and the convergence point is in x;

Question conditions: Cantor's closed-set theorem.

It can be proved by following the method of mathematical analysis.

Definition A compact set is a special set of points in a topological space that has finite sub-overrides for any open override. Within a measure space, a tight set can also be defined as a set that satisfies any of the following criteria:

Any column has a converging subcolumn, and the limit points of the subcolumn belong to that set (self-compaction set).

It has the properties of Bolzano-Weierstrass.

Complete and fully bounded.

Properties Compact sets have the following properties:

A tight set is necessarily a bounded closed set, but the reverse is not necessarily true.

A compact set under a continuous function is still a compact set.

The compact subset of the Hausdorf space is the closed set.

The non-empty compact subset of the real space has the largest and smallest elements.

Heine-Borel theorem: Within RN, a set is compact if and only if it is closed and bounded.

The continuous real-valued function defined on a tight set is bounded and has a maximum and minimum value.

The continuous real-valued function defined on the compact set is consistently continuous.

Intuitive understanding. In a sense, a compact set is similar to a finite set. To take the simplest example, in a metric space, all finite sets have the largest and smallest elements. In general, an infinite set may not have a maximum or minimum element (e.g., (0,1) in r), but a non-empty compact subset in r has both a maximum and a minimum element.

In many cases, a proof of true for a finite set can be extended to a tight set. A simple example is a proof of the property that a continuous real-valued function defined on a compact set is uniformly continuous.

Similar concept. Self-containment compact: Each bounded sequence has convergent subsequences.

Countable compact set: Each countable open cover has a finite sub-cover.

Pseudo-tightness: All real-valued continuous functions are bounded.

Weakly countable compaction: Each infinite subset has a limit point.

In the metric space, the above concepts are all equivalent to compact sets.

The following concepts are generally weaker than compact sets:

Relative compactness: If a subspace y closure in the parent space x is compact, then y is said to be relatively compact to x

Quasi-compact set: If all sequences in space x's subspace y have a convergent subsequence, then y is said to be a quasi-compact set in x.

Local compact space: If each point in the space has a local base consisting of compact neighbors, the space is said to be a local compact space.