Limit point

In calculus, an accumulation point of a set is clearly a point that has infinitely many points of the set in its vicinity. An accumulation point of a sequence ( rare: " compression point" or " accumulation value") is a point which is the limit of a subsequence. Both terms are closely related. Appropriate, but in detail there are slightly different definitions in the topology. The concept of accumulation point plays an important role in mathematics.

A stronger condition is true for a condensation point or - cluster point (see below) of a set.

• 4.1 Definition
• 4.2 Example

Accumulation points and limits

The term consequence of accumulation point is closely related to the term limit. The crucial difference is that each sequence can have a maximum limit, but possibly several, perhaps even an infinite number of accumulation points.

From a limit is requested that are almost all followers in any environment. In a cluster point, this must only be infinitely many. It can therefore again "infinitely many" followers for further accumulation points remain. If a sequence has a limit, then this limit is especially a result of an accumulation point. If a sequence in a Hausdorff space ( ie, in particular, each sequence in a metric space ) has several follow- cluster points, then it has no limit.

Consequence of accumulation points and amount of accumulation points

The terms sequence accumulation point and quantity accumulation point are closely related, but not exactly equivalent. This is demonstrated in the following example:

The result is defined as follows:

The sequence a has two accumulation points. The subsequence converges to 0, so 0 is an accumulation point of sequence. The subsequence converges to 1, that is also an accumulation point of sequence 1.

The amount of followers of is defined by

That is, the set of all follower members, see Fig set of functions. Now 0 is an accumulation point of ε for every environment, there are still elements with which one does not, however, as there is for example in its environment with the radius no other element of the set.

The difference is due to the fact that a value infinitely often occurs as a term in a sequence, is still counted only once in the set. Each quantity accumulation point is a consequence of an accumulation point. Conversely, a result of an accumulation point, either a quantity limit point, or infinitely often comes before and follower.

Accumulation point of a sequence

Definition

This definition is initially valid for sequences of rational or real numbers. You can literally just as in any, even multi-dimensional metric spaces, are more commonly used still in uniform spaces and beyond in all topological spaces. Here, a more general definition of each term environment is used.

If the topology of the space too ' lumpy ' is not a point is an accumulation point, if in any environment is of one of various follower.

A sequence may have one, several, even infinitely many accumulation points, between which they in their course " jumps back and forth ." Similarly, there are consequences that have no accumulation point.

In a compact space has every infinite sequence has an accumulation point (for example, in a bounded and compact subset of the real space ).

Accumulation points and limits

The limit of a convergent sequence is always an accumulation point of the sequence, since by definition each of the limit, however small environment contains all but finitely many terms of the sequence. In metric spaces, and more generally in Hausdorff spaces, the limit of a convergent sequence is unique and is the only accumulation point of the sequence. In general topological spaces, a sequence simultaneously possess both a limit and a limit point which is not a limit.

Subsequences

Has a consequence a limit, converge all subsequences against this. For a cluster point, it is sufficient that a subsequence converges to the accumulation point. Every accumulation point of a subsequence is also an accumulation point of the output sequence. In the space of real numbers (and more generally in all the first axiom of countability fulfilling topological spaces ), for every limit point of a subsequence that converges to this accumulation point.

Limit superior and limit inferior

For limited real number sequences, the limit superior is defined ( in German "upper limit " or " upper limit " ) as the largest accumulation point. It exists because of the amount of accumulation points is non-empty, closed and bounded. You write it.

Where: is the largest accumulation point of a sequence if and only if for each in the interval infinitely many, but are at most finitely many terms of the sequence to be found in the interval.

Similarly, the limit inferior is defined as the smallest cluster point of a bounded real number sequence. It is true.

Limit superior and limit inferior can be generalized to the extended real numbers and then close the value and consequences for unlimited down as accumulation points for unrestricted upward sequences a. In order to distinguish and often referred to in this context as an improper accumulation points. Including the improper accumulation points exist Limes superior and limes inferior then not only limited, but also to all arbitrary real number sequences.

Examples

• The constant real-valued sequence has 1 as its only accumulation point. The elements of the sequence jump between 1 and -1 back and forth, and both points are accumulation points of the sequence, although there are, for example environments by 1, so that infinitely many terms of the sequence lie outside the area. At the same time, the converging subsequence of the elements with a straight sequence index for the upper limit point 1, and the subsequence of elements with odd index sequence converges to the lower limit point -1.

• The sequence converges to 0, and 0 is therefore the only accumulation point of the sequence. The example shows that the point of accumulation of the result itself does not need to occur in the sequence.

• The real-valued divergent sequence has no accumulation point. By adding a " point at infinity " ( Einpunktkompaktifizierung ) can be the set of real numbers into a compact space to expand, where is the added point the only accumulation point of the sequence.

• In one provided with the indiscrete topology space, each point of the room limit point and even limit each episode: the indiscrete topology is the gröbstmögliche topology, and in such a room the whole room itself is the only non-empty open set and thus the only than ambient question coming crowd. In one provided with the discrete topology space, however, is a point if and only accumulation point of a sequence when infinitely often appears as an element of the result: the discrete topology is the finest possible topology, and in such a space are also open to the singleton subsets. Thus any one-element subset is the smallest neighborhood of the point contained in it.

Accumulation points and tangency an amount

Definition

In a topological space is a point of the base amount and a subset of. This is known as the contact point (also Adhärenzpunkt ) of when located in any environment from at least one point of. is called an accumulation point of if is in any environment from at least one point of which is different from. The set of all accumulation points of a set is called a derivation of the crowd. The set of all points of contact by means of completion and is written as.

In topological spaces is:

• Just then a limit point of if,
• Every accumulation point is a point of contact,
• Each point is a point of contact,
• Each contact point, which lies in, also a limit point of.

In this context, called an accumulation point of proper ( or proper accumulation point ) if every neighborhood has an infinite number of points in common with.

In a T1 - space are the concepts of limit point and limit point in the strict sense equivalent, and each point is under the condition that the neighborhood filter of each point of the space has a countable base iff a limit point of if there is a of points of \ existing impact is that converges to.

Be the neighborhood filter of the point in the topological space. This is called

The degree of compaction of the amount in point. For each cardinal number is called an - limit point of if. The - cluster points are called maximal or complete accumulation points. The - cluster points (read Aleph -1 cluster points ) are called compression or condensation points. The set of all points in the condensing points of a set are called condensation of and is referred to or not. In Polish spaces applies to any amount.

Is called isolated point of when he is in, but is not a limit point of. ie uncompressed if it is not a point of compression. Quantities without isolated points are called insichdicht. Quantities, which consist only of isolated points, hot isolated quantities. In a T1 - space, the closure of a insichdichten quantity as well as the union of insichdichten quantities are insichdicht. The relatively open subsets of a insichdichten amount are also insichdicht. The union of all insichdichten subsets of is called the core of insichdichte. Sets whose insichdichte cores are empty hot, separated. Each isolated amount is separated, but not vice versa. In a T1 - space of insichdichte core is greatest of respect to the inclusion insichdichte subset of. Completed insichdichte sets are called perfect. In Polish spaces a lot is perfect if and only if.

Example

Let be a subset of the real numbers. therefore consists of a left half-open interval and a single point. With the exception of all elements of cluster points of. The isolated, for example because the open interval is a neighborhood of which contains no other point.

In addition, the accumulation point of zero. Because the interval is left open, there are points in the interval, which are arbitrarily close to zero. Thus every neighborhood of the root must contain a point of the interval. For the same reason it is also the limit point of. Here it is clear that a limit point can belong on the quantity, but not necessarily.

Other names

Sometimes the words limiting point, point or boundary point be used instead of an accumulation point.