Cluster point

In analysis , an accumulation point of a set is clearly a point that has an infinite number of points of the set in its vicinity. An accumulation point of a sequence (more rarely: "condensation point" or "accumulation value") is a point that is the limit value of an infinite partial sequence . Both terms are closely related. Corresponding definitions that are slightly different in detail can be found in the topology . The concept of the cluster point plays an important role in mathematics .

A stronger condition applies to a point of condensation or a point of accumulation (see below) of a set. ${\displaystyle \beth _{1}}$

Accumulation points and limit values

The term cluster point is closely related to the term limit value . The crucial difference is that each sequence can have at most one limit value, but possibly several , perhaps even an infinite number of accumulation points.

It is required of a limit value that almost all elements of the sequence are in every environment . In the case of an accumulation point , this only has to be infinite . So an “infinitely many” sequence elements can remain for further accumulation points. If a sequence has a limit value, then this limit value is in particular also a sequence accumulation point. If a sequence in a Hausdorff space (especially every sequence in a metric space ) has several sequence accumulation points, then it has no limit value.

Sequence accumulation points and quantity accumulation points

The terms sequence clustering point and set clustering point are closely related but not exactly equivalent. The following example demonstrates:

The constant subsequence converges to 1, the nontrivial others to 0.

The sequence is defined as follows: ${\displaystyle a=(a_{n}),n\in \mathbb {N} }$

${\displaystyle a_{n}:={\begin{cases}1/n,&{\text{falls }}n{\text{ gerade}}\ ,\\1,&{\text{sonst.}}\end{cases}}}$

The result has two accumulation points. The subsequence converges to 0, so 0 is the sequence accumulation point of . The subsequence converges to 1, so 1 is also a sequence accumulation point of . ${\displaystyle a}$ ${\displaystyle (a_{2n})}$ ${\displaystyle (a_{n})}$ ${\displaystyle (a_{2n+1})}$ ${\displaystyle (a_{n})}$

The only accumulation point of the set is 0, and 0 itself does not belong to the set. 1 is not a cluster point.

The set of terms of is defined by ${\displaystyle (a_{n})}$

${\displaystyle X((a_{n})):=\left\{a_{n}\mid n\in \mathbb {N} \right\}}$

That is, is the set of all sequence members, see the image set of functions. Now 0 is an accumulation point of the set because around every -surrounding there are still elements with , but not the 1, because for example there is no further element of the set in its vicinity with the radius . ${\displaystyle X((a_{n}))\,}$ ${\displaystyle X((a_{n})),}$ ${\displaystyle \varepsilon }$ ${\displaystyle 1/n\,}$ ${\displaystyle 1/n<\varepsilon }$ ${\displaystyle 0{,}3}$

The difference is based on the fact that a value that occurs infinitely often as a member in a sequence is nevertheless only counted once in the set. Each set cluster point is a sequence cluster point. Conversely, a sequence cluster point is a set cluster point or occurs infinitely often as a sequence member.

Accumulation point of a sequence

definition

A point is called an accumulation point or accumulation value of a sequence of points if there are infinitely many elements of the sequence in every little neighborhood of the point. ${\displaystyle p}$

This definition initially applies to sequences of rational or real numbers . It can literally also be used in any, including multidimensional, metric spaces , more generally in uniform spaces and, moreover, in all topological spaces . A more general definition of the concept of environment is used in each case.

Unless the topology of space is not 'clumped' too, a point is already then a cluster point if every neighborhood of one of is different follower. ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

A sequence can have one, several, even an infinite number of accumulation points, between which it "jumps back and forth" in its course. There are also consequences that do not have an accumulation point.

In a compact space , every infinite sequence has an accumulation point (for example in a restricted and closed part of real space).

Accumulation points and limit values

The limit value of a convergent sequence is always also the accumulation point of the sequence, because by definition every area around the limit value, however small, contains all but a finite number of sequence elements. In metric spaces and more generally in Hausdorff spaces , the limit value of a convergent sequence is unambiguous and is also the only accumulation point of the sequence. In more general topological spaces, a sequence can simultaneously have both a limit value and an accumulation point that is not a limit value.

Partial consequences

If a sequence has a limit value, all partial sequences converge towards this. For an accumulation point it is sufficient that a subsequence converges towards the accumulation point . Each accumulation point of a partial sequence is also an accumulation point of the initial sequence. In the space of real numbers (and more generally in all topological spaces that satisfy the first axiom of countability ) there is a subsequence for each accumulation point which converges to this accumulation point.

Limes superior and limes inferior

Provided that the set of accumulation points of a bounded real number sequence is not empty , restricted and also closed , the Limes superior (in German "upper Limes" or "upper limit value") is defined as the largest accumulation point of this sequence, written . ${\displaystyle \textstyle \limsup _{n\to \infty }a_{n}}$

The following applies: is the largest accumulation point of a sequence if and only if there are infinitely many (further) sequence members for each in the interval , but at most finitely many (further) sequence members in the subsequent interval . ${\displaystyle b\;}$ ${\displaystyle a=(a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \varepsilon >0\;}$ ${\displaystyle (b-\varepsilon ,b+\varepsilon )\;}$ ${\displaystyle (b+\varepsilon ,\infty )}$

Similarly, the Limes inferior (in German “lower Limes” or “lower limit value”) is defined as the smallest accumulation point of a restricted real number sequence. It applies . ${\displaystyle \textstyle \liminf _{n\to \infty }a_{n}=-\limsup _{n\to \infty }(-a_{n})}$

Limes superior and Limes inferior can be generalized to the extended real numbers and then include the value for sequences that are unbounded above and as accumulation points for sequences that are unbounded below . To distinguish between, and in this context are often referred to as improper accumulation points. Including the improper accumulation points, limes superior and limes inferior then exist not only for restricted, but for any real number sequences. ${\displaystyle +\infty }$ ${\displaystyle -\infty }$ ${\displaystyle +\infty }$ ${\displaystyle -\infty }$

Examples

The result has two accumulation points ${\displaystyle (-1)^{n}\cdot {\frac {n}{n+1}}}$

• The constant real-valued sequence has 1 as the only accumulation point. The elements of the sequence jump back and forth between +1 and −1, and both points are accumulation points of the sequence, although there are neighborhoods around +1, for example, so that an infinite number of sequence members lie outside the neighborhood. At the same time, the sub-sequence of elements with an even sequence index converges towards the upper accumulation point +1, and the sub-sequence of elements with an odd sequence index converges towards the lower accumulation point −1. ${\displaystyle a_{n}=1}$ ${\displaystyle b_{n}=(-1)^{n}}$
• The sequence converges to 0, and 0 is accordingly the only accumulation point of the sequence. The example shows that the accumulation point of the sequence itself does not need to appear in the sequence. ${\displaystyle c_{n}=1/n}$
• The real-valued divergent sequence has no accumulation point. By adding a “point at infinity” ( one-point compactification ), the set of real numbers can be expanded into a compact space in which the added point is the only accumulation point in the sequence. ${\displaystyle d_{n}=n}$
• An example of a sequence with a countably infinite number of accumulation points is . Every natural number is an accumulation point of this sequence. ${\displaystyle (a_{n})=(1,1,2,1,2,3,1,2,3,4,\dots )}$
• There are also sequences with an uncountable infinite number of accumulation points. Since the rational numbers are countable, there is a bijection . The construction is based on Cantor's first diagonal argument . This bijection can now be understood as a sequence in the real numbers. Since in lies close , every real number is an accumulation point of this sequence. ${\displaystyle \phi \colon \mathbb {N} \to \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$
• In a space provided with the indiscreet topology , every point of the space is an accumulation point and even the limit value of every sequence: The indiscreet topology is the coarsest possible topology, and in such a space the entire space itself is the only non-empty open set and therefore the only one in question as surroundings coming crowd. In a space provided with the discrete topology , on the other hand, a point is the point of accumulation of a sequence if and only if it appears infinitely often as an element of the sequence: The discrete topology is the finest possible topology, and in such a space the single-element subsets are also open. Thus every single element subset is the smallest possible neighborhood of the point it contains.
• The sequence has 1 and −1 as the accumulation point. This can also be seen well in the following graphic, which shows some elements of this sequence: ${\displaystyle a_{n}=(-1)^{n}{\tfrac {n}{n+1}}}$

Points of accumulation and points of contact of a crowd

definition

In a topological space be a point of the basic set and a subset of . It is called the point of contact (also point of adherence ) of when there is at least one point of in every environment of . is called an accumulation point of if in every neighborhood of there is at least one point of which is different from . One can thus characterize set accumulation points in such a way that they can be approximated with arbitrary precision by other elements of the set. The set of all accumulation points of a set is called the derivative of the set . The set of all points of contact is called the closure of and is written as. ${\displaystyle (X,T)}$ ${\displaystyle p}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle {\overline {M}}}$

In topological spaces:

• ${\displaystyle \displaystyle p}$an accumulation point of if and only if , ${\displaystyle \displaystyle M}$ ${\displaystyle p\in {\overline {M\setminus \{p\}}}}$
• each accumulation point a point of contact,
• every point a point of contact, ${\displaystyle p\in M}$
• every point of contact that lies in is also an accumulation point of . ${\displaystyle \displaystyle X\setminus M}$ ${\displaystyle \displaystyle M}$

In this context, the point of accumulation is called in the narrower sense (or the actual point of accumulation ), if every neighborhood of has an infinite number of points in common . ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$

In a T ₁ space, the terms cluster point and cluster point are equivalent in the narrower sense , and each point is, provided that the environment filter of each point in the space has a base that can be counted at most , a cluster point of if and only if it is one of points of existing sequence there that converges against . ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle M\setminus \{p\}}$ ${\displaystyle p}$

Let be the environment filter of the point in topological space . Is called ${\displaystyle B_{X}(p)}$ ${\displaystyle p}$ ${\displaystyle X}$

${\displaystyle \operatorname {v} (X,M,p)=\min\{\operatorname {card} (M\cap U)\}_{U\in B_{X}(p)}}$

the degree of compression of the quantity in the point . For every cardinal number is one - an accumulation point of when . The accumulation points are called maximum or complete accumulation points . The accumulation points (read Beth-1 accumulation points) are called compression or condensation points . The set of all points in that are condensation points of a set is called condensation of and is denoted by or . In Polish areas, the same applies to a lot . ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle m\leq \operatorname {card} (X)}$ ${\displaystyle p}$ ${\displaystyle m}$ ${\displaystyle M}$ ${\displaystyle m\leq \operatorname {v} (X,M,p)}$ ${\displaystyle \operatorname {card} (X)}$ ${\displaystyle \beth _{1}}$ ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \operatorname {cp} (M)}$ ${\displaystyle M^{\odot }}$ ${\displaystyle M\colon \operatorname {cp} (\operatorname {cp} (M))=\operatorname {cp} (M)}$

${\displaystyle p}$is called an isolated point of if it lies in, but is not an accumulation point of . is called uncompressed if it is not a compression point of . Sets without isolated points are called inself-dense . Sets that consist only of isolated points are called isolated sets . In a T ₁ space, the closed envelope of an in-itself set and the union of in-itself sets are in-itself-tight. The relatively open subsets of an in-itself dense set are also in-itself dense. The union of all inself-dense subsets of is called the inself-dense kernel of . Sets whose insecurely dense nuclei are empty are called separated . Every isolated set is separated, but not vice versa. In a T ₁ space, the inself-dense core of is the largest inself-dense subset of in terms of inclusion . Closed, insecure sets are called perfect . In Polish spaces, a lot is perfect precisely when . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \operatorname {cp} (M)=M}$

example

Let be a subset of the real numbers . thus consists of a left half-open interval and a single point. With the exception of , all elements of are accumulation points of . This is isolated because, for example, the open interval is a neighborhood of that contains no further point from . However, it is a touchpoint of the crowd, because it lies in each of its surroundings and is an element of the crowd . ${\displaystyle M:=(0,1]\cup \{3\}}$ ${\displaystyle M}$ ${\displaystyle 3}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle 3}$ ${\displaystyle (2,4)}$ ${\displaystyle 3}$ ${\displaystyle M}$ ${\displaystyle 3}$ ${\displaystyle M}$

In addition, there is also the zero accumulation point of . Since the interval is open on the left, there are points in the interval that are arbitrarily close to zero. Thus, every neighborhood of zero must also contain a point in the interval. For the same reason is also the accumulation point of . Here it becomes clear that an accumulation point can belong to the set , but does not have to. ${\displaystyle M}$ ${\displaystyle 1}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$

Other names

Sometimes will take a limit point the words limiting point , - point or limit point used. ${\displaystyle \beta }$

Web links

Wikibooks: Math for non-freaks: accumulation point of an episode  - learning and teaching materials

Individual evidence

1. ^ Boto von Querenburg : Set theoretical topology. Springer-Verlag, 2001, ISBN 3-540-67790-9 .
2. ↑ ^{a b} Lynn Arthur Steen, J. Arthur Seebach, Jr .: Counterexamples in Topology. Dover Publications, 1995, ISBN 0-486-68735-X .
3. ↑ Harro Heuser: Textbook of Analysis. Part 1, Chapter 29: Improper limit values, accumulation values ​​and limits
4. ↑ ^{a b c d e f g} J. Naas, HL Schmid: Mathematical dictionary. BG Teubner, Stuttgart 1979, ISBN 3-519-02400-4 .
5. ↑ ^{a b c} W. Rinow: Textbook of Topology. VEB Deutscher Verlag der Wissenschaften, 1975, DNB 760148066 .
6. ^ K. Kuratowski: Introduction to Set Theory and Topology. Elsevier Science, 1972, ISBN 0-08-016160-X .
7. ↑ ^{a b} F. Hausdorff : Fundamentals of set theory. 1914. Chelsea Publishing Company, New York 1949, chap. VII