拓扑 稠密集和无处稠密集 Dense Set

01

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稠密集 Dense Set

在拓扑学及数学的其它相关领域，给定拓扑空间X及其子集A，如果对于X中任一点x，x的任一邻域同A的交集不为空，则A称为在X中稠密。直观上，如果X中的任一点x可以被A中的点很好地逼近，则称A在X中稠密。

In topology and other related fields of mathematics, A given topological space X and its subset A is said to be dense in X if, for any point x in X, the intersection of any neighborhood of x with A is not empty. Intuitively, A is said to bedensein X if any point x in X is well approximated by any point in A.

等价地说，A在X中稠密当且仅当X中唯一包含A的闭集是X自己。或者说，A的闭包是X，又或者A的补集的内部是空集。

Equivalently, A is dense in X if and only if the only closed set in X that contains A is X itself.In other words, the closure of A is X, or the interior of the complement of A is an empty set.

1.1 定义 Definition

在度量空间(E,d)中，也可以如下定义稠密集。当X的拓扑由一个度量给定时，在X中A的闭包是A与A中元素的所有数列极限（它的极限点）的集合的并集，

In the metric space (E,d), a dense set can also be defined as follows.When the topology of X is given by A metric, the closure of A in X is the union of A with the set of all sequence limits (its limit points) of the elements in A,

那么当

When

A在X中是稠密的。即A在X中稠密当且仅当A的闭包是X。

A is dense in X.That A is dense in X if and only if the closure of A is X.

注意。如果是一个完备度量空间X中稠密开集上的序列，则在X上依然稠密。这个事实与贝尔纲定理中的一个形式等价。

Pay attention to. Ifis a sequence on a dense open set in a complete metric space X,is still dense on X. This fact is equivalent to a form ofBaire category theoremBaire category theorem.

1.2性质 properties

(1) A在X中稠密的充要条件是，对于任意一个x∈X，在x的任何邻域内都有A的点。这条性质有时也被作为稠密集的另一种定义。

The necessary and sufficient condition for A to be dense in X is that for any x ∈X, there are points of A in any neighborhood of X.This property is sometimes used as another definition of a dense set.

必要性：因为A在X中稠密，所以，所以。于是根据闭包的性质，x的任何邻域与A的交集都不空，即x的任何邻域内都有A的点。反过来推导即可证明充分性。

Necessity: because A is dense in X, so, so.So by the nature of closures, the intersection of any neighborhood of x with A is not empty, that is, any neighborhood of x has A point of A.The proof of adequacy can be derived in reverse.

(2) 若A在X中稠密，则对于任意一个x∈X，在A中都能找到一个点列，使得收敛于x。

If A is dense in X, then for any X ∈X, A point column can be found in A so that converges to X.

证明：，即x要么是A的内点，要么是A的边界点。若x是内点，因为内点必是聚点，于是根据聚点的定义，存在某个各项互异的点列，使得收敛于x，命题得证。若x是边界点，因为边界点可能是聚点，也可能是孤立点，又分为两种情况。若x是边界点中的聚点，则命题得证。若x是边界点中的孤立点，由孤立点的定义，x∈A，于是可取常数列，而常数列总是收敛的。

Proof:, that x is either the interior point of A or the boundary point of A. If x is the interior point, because the interior point must be the convergence point, then according to the definition of the convergence point, there exists some mutually different point column , so that converges to x, the proposition is proved.

If x is the boundary point, because the boundary point may be a cluster point, or it may be an isolated point, there are two cases. If x is a convergence of boundary points, the proposition is proved. If x is an isolated point in the boundary point, then by the definition of an isolated point, x∈A, the constant sequence can be taken, and the constant sequence always converges.

1.3 例子 Examples

每一拓扑空间是其自身的稠密集。

Each topological space is its own dense set.

有理数域和无理数域是实数域中的稠密集（在通常拓扑意义下）。

The rational number field and the irrational number field are dense sets (in the usual topological sense) of the real number field.

度量空间M是其完备集γM中的稠密集

The metric space M is a dense set of its complete set γM

02

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无处稠密集 Nowhere Dense Set

在拓扑学中，一个拓扑空间X的子集A称为无处稠密集，如果A的闭包的内部是空集。例如，整数在实数轴R上就形成了一个无处稠密集。

In topology, A subset of topological space X, A, is called A nowhere dense set if the interior of A's closure is an empty set.For example, integers form a nowhere dense set on the real number line R.

2.1 定义 Definition

以下定义等价。

The following definitions are equivalent.

（1）拓扑空间(X,τ)，A⊆X，称A是无处稠密的（亦称稀疏的，或称A为无处稠密集、稀疏集，疏朗集），当且仅当A的闭包的内部是空集。或者说，A是无处稠密集，当且仅当A的闭包不含内点，或A的闭包不包含任何邻域。

Topological space (X,τ), A ⊆ X, said A isnowhere dense(also known assparse,or call Anowhere dense set,sparse set), if and only if the closure of A set of inside is empty.In other words, A is A nowhere dense set if and only if the closure of A contains no inner points, or the closure of A contains no neighborhood.

（2）如果A不在X的任何一个非空开子集中稠密，就称A是无处稠密集。即对于X的任意一个非空开子集E，A在E中都不稠密。对于X的任意一个非空开子集E，A在E中稠密即是指对任意x∈E以及任意ε>0，邻域B(x,ε)中都有A的点。因此无处稠密的意思就是指，对于X的任意一个非空开子集E，存在某个x0∈E以及某个ε0>0，使得邻域B(x0,ε0)中没有A的点。

A is said to be A nowhere dense set if A is not dense in any nonempty open set of X. So for any non-open subset of X, E, A is not dense in E.For any non-open subset E of X, A is dense in E, that is, for any X ∈E and anyε>0, the neighborhood B(x,) has A.

So nowhere dense means that for any nonempty open subset E of X, there exists some x0∈E and someε0>0, so that there is no A in the neighborhood of B(x0, 0).

注意，“对于X的任意一个非空开子集E”这个条件是必要的，即A必须在X的任何一个开子集中都不稠密，才能被称作无处稠密。如果只是某些开子集中不稠密，而在另一些开子集中稠密，这不叫做无处稠密集。

Note that the condition "for any non-open subset E of X" is necessary, that A must be not dense in any open set of X in order to be called nowhere dense.If you just have some open sets that are not dense, and some open sets that are dense, this is not called a nowhere dense set.

证明：（1）推（2），设A的闭包不含内点，那么对X的任意一个非空开子集E，集合。这是因为假设，那么。然而E是非空开集，开集中的每个点都是内点，即A的闭包中每个点都是内点，这与A的闭包不含内点矛盾。既然，而开集与闭集（任一集合的闭包一定是闭集）的差集仍是开集，可知是开集。根据U是开集的事实可知，存在某个x0∈U⊆E以及某个ε0>0，邻域B(x0,ε0)⊆U。再根据差集的定义，U中不含的点，因此B(x0,ε0)也不含的点。然而，不含的点即意味着不含A的点，结合E的任意性即可得到：对于X的任意一个非空开子集E，存在某个x0∈E以及某个ε0>0，使得邻域B(x0,ε0)中没有A的点。

Proof :(1) deduces (2), if the closure of A does not contain inner points, then for any non-empty open subset E of X, set.That's because of the hypothesis, so. However, E is A non-empty open set, and every point in the open set is an inside point, that is, every point in the closure of A is an inside point, which contradicts the fact that the closure of A does not contain an inside point.

Since, the difference between an open set and a closed set (the closure of any set must be a closed set) is still an open set, we know thatis an open set. According to the U is the fact that open set, there is a certain x0 ∈ U ⊆ E and a epsilon zero &ε0>0, neighborhoodB(x0,ε0)⊆U.

And by the definition of the difference set, that U does not contain the points in, soB(x0,ε0)does not containthe points in.

However, the points not containsmeans the points not contains A, for any non-empty open subset E of X, yields: for any nonempty open subset E of X, there exists an x0∈E and anε0>0, so that there is no A in the neighborhood of B(x0,ε0).

（2）推（1），反证法，假设A的闭包含有内点，或A的闭包包含了某个邻域E，此时E中任意一点都属于A的闭包。根据闭包的定义，A的闭包中的点均满足：任意一点的任何一个邻域与A的交集都非空。因为E中任意的点都属于A的闭包，所以E中任意一点的任意一个邻域与A的交集非空，这就表示A在E中稠密。然而，这与A是无处稠密集矛盾。

(2) deduce (1)， proved by contradiction, suppose the closure of A contains an inner point, or the closure of A contains some neighborhood E, then any point in E belongs to the closure of A. By the definition of A closure, the points in A's closure satisfy that any neighborhood of any point has A non-empty intersection with A.

Since any point in E is A closure of A, the intersection of any neighborhood of any point in E and A is not empty, which means that A is dense in E. However, this contradicts that A is A nowhere dense set.

2.2 性质 Properties

（1）若E在X中无处稠密，则也在X中无处稠密，反之亦然。

If E is nowhere dense in X, thenis nowhere dense in X, and vice versa.

这是因为若E是无处稠密集，则无内点。因为是闭集，所以

，所以无内点。而无内点就意味着是无处稠密集。反之亦然。

This is because if E is a nowhere dense set,has no interior point. Becauseis a closed set, so,has no inner point. Andhas no interior points meansis nowhere dense set. And vice versa.

（2）E在X中无处稠密的充要条件是在X中稠密。

The necessary and sufficient condition for E to be nowhere dense in X is thatis dense in X.

必要性：设E在X中无处稠密，，要证明U在X中稠密，只需要证明。根据闭包和内部的对偶性，，因此U在X中稠密，即在X中稠密。

Necessity: suppose that E is nowhere dense in X,, to prove that U is dense in X, one only needs to prove. According to the duality of the closure and the interior,, U is therefore dense in X, that is,is dense in X.

充分性：设在X中稠密，，则。两边取补集，，即E的闭包不含内点，因此E在X中无处稠密。

Sufficiency: setthe density in X,, then. Take the complement of both sides,, that is, the closure of E has no inner point, so E is nowhere dense in X.

注意，E在X中无处稠密也可推出Ec在X中稠密，按照必要性的推理过程，要证Ec稠密，只要证，亦即。若假设E含内点，则存在内点p的某个邻域B(p)，使得，即含有邻域，这与E是无处稠密集矛盾。因此，即Ec在X中稠密。这也就是说，无处稠密集的补集一定是稠密集。

Notice that E is nowhere dense in X andEcis dense in X, according to the necessary reasoning process, to proveEcis dense, as long as the proof, namely.

If E is assumed to contain the inner point, then there is some neighborhood B(p) of the inner point p, so that, namely,contains the neighborhood, which contradicts that E is a nowhere dense set. So, namelyEcis dense in X. That is, the complement of a dense set everywhere must be a dense set.

但稠密集的补集不一定是无处稠密集，例如有理数集Q在R中稠密，但有理数集的补集，即无理数集并不无处稠密，事实上无理数集也是个稠密集。

But the complement of a dense set is not necessarily a nowhere dense set. For example, Q of rational Numbers is dense in R, but the complement of a rational number set is not nowhere dense. In fact, the set of irrational Numbers is also a dense set.

3.3 例子 Examples

例如，整数在实数轴R上就形成了一个无处稠密集。这是因为数轴上任何一个开区间，只要取某个子区间落在任意两个整数之间，那么这个子区间不包含任何整数点，因此整数集Z在R内无处稠密。

For example, integers form a nowhere dense set on the real number lineR. This is because any open interval on the number line, as long as you take some subinterval between any two integers, that subinterval does not contain any integer points, so the set of integers Z is nowhere dense inR.

注意运算的次序是很重要的，内部的闭包为空，不代表闭包的内部也为空。例如，有理数集Q，它没有内点，因为无理数集的稠密性，不存在这样的有理数p，使得p的某个邻域上全部是有理数，因此Q的内部是空集。而空集是闭集，Q的内部的闭包（注意不是“闭包的内部”）就是空集本身。但Q不是无处稠密集，实际上，它在R上是稠密的，正好相反。

It is important to note the order of operations. Just because the inner closure is empty does not mean that the inner closure is also empty.

For example, the rational number set Q, it has no interior point, because the density of the irrational number set, there is no such rational number p, so that all rational Numbers on some neighborhood of p are rational Numbers, so the interior of Q is an empty set. While an empty set is a closed set, the inner closure of Q (note that it is not the "inside of a closure") is the empty set itself.

But Q is not a dense set anywhere, in fact, it's dense on R, just the opposite.

无处稠密与周围的空间也有关：有可能把一个集合考虑为X的子空间时就是无处稠密的，但考虑为Y的子空间时，就不是无处稠密的。显然，一个集合在它本身中总是稠密的。

Nowhere dense is also related to the surrounding space: it is possible to think of a set as a subspace of X that is nowhere dense, but not as a subspace of Y that is nowhere dense.Obviously, a set is always dense in itself.

开集和闭集 Open set and closed set

一个无处稠密集不一定是闭集（例如，集合在实数集上是无处稠密集），但一定是包含在一个无处稠密的闭集（即它的闭包）内。确实，一个集合是无处稠密集，当且仅当它的闭包是无处稠密集。

A nowhere dense set is not necessarily a closed set (for example, a setis nowhere dense on a set of real Numbers), but it must be contained in a nowhere dense closed set (that is, its closure).Indeed, a set is a nowhere dense set if and only if its closure is a nowhere dense set.

无处稠密的闭集的补集是一个稠密的开集，因此无处稠密集的补集是内部为稠密的集合。

The complement of a closed set with nowhere dense is a dense open set, so the complement of a nowhere dense set is an internally dense set.

测度为正数 Positive Measure

一个无处稠密集并不一定就是可忽略的。例如，如果X位于单位区间[0,1]，不仅有可能有勒贝格测度为零的稠密集（例如有理数集），也有可能有测度为正数的无处稠密集。

A nowhere dense set is not necessarily negligible. For example, if X is in the unit interval [0,1], it is possible not only to have a dense set with a lebesgue measure of zero (such as a set of rational Numbers), but also to have a nowhere dense set with a positive measure.

例如（一个康托尔集的变体），从[0,1]内移除所有形为a/2n的最简二进分数，以及旁边的区间[a/2n−1/22n+1,a/2n+1/22n+1]；由于对于每一个n，这最多移除了总和为1/22n+1的区间，留下的无处稠密集的测度就至少是1/2（实际上刚刚大于0.535……，因为重叠的原因），因此在某种意义上表示了[0,1]的大多数空间。

For example (a variant of the cantor set), remove from [0,1] the simplest binary fraction visible as[a/2n−1/22n+1,a/2n+1/22n+1]; Since for each n this removes, at most, an interval whose sum is 1/22n+1, the measure of the remaining nowhere dense set is at least 1/2 (actually just over 0.535..., because of the overlap), so it represents most of the space [0,1] in a sense.

把这个方法进行推广，我们可以在单位区间内构造出任意测度小于1的无处稠密集。

By extension of this method, we can construct a nowhere dense set with any measure less than 1 in the unit interval.