What Is The Derived Set Of A Set A In A Topological Space?

The derived set of a set Acap A 𝐴 in a topological space is the set of all its limit points (also known as accumulation points or cluster points). It is denoted by A′cap A prime

𝐴′

or D(A)cap D open paren cap A close paren

𝐷(𝐴)

. A point xx

𝑥

in a topological space Xcap X

𝑋

is a limit point of Acap A

𝐴

if every open neighborhood of xx

𝑥

contains at least one point of Acap A

𝐴

that is different from xx

𝑥

.

The derived set captures the essential "boundary" or "accumulation" behavior of a set within a given topology. Unlike the closure, which includes all points of the original set, the derived set consists exclusively of points that can be "approached" by other points of the set.

Detailed definition and notation

Let (X,τ)open paren cap X comma tau close paren

(𝑋,𝜏)

be a topological space, where Xcap X

𝑋

is the set and τtau

𝜏

is the collection of open sets. Let Acap A

𝐴

be a subset of Xcap X

𝑋

.A point x∈Xx is an element of cap X

𝑥∈𝑋

is a limit point of Acap A

𝐴

if for every open set U∈τcap U is an element of tau

𝑈∈𝜏

such that x∈Ux is an element of cap U

𝑥∈𝑈

, we have:(U∖{x})∩A≠∅open paren cap U ∖ the set x end-set close paren intersection cap A is not equal to the empty set

(𝑈∖{𝑥})∩𝐴≠∅

The derived set of Acap A

𝐴

, denoted A′cap A prime

𝐴′

, is the set of all such limit points.

The key part of this definition is the phrase "other than xx

𝑥

itself." This condition distinguishes limit points from isolated points. An isolated point of a set Acap A

𝐴

is a point a∈Aa is an element of cap A

𝑎∈𝐴

for which there exists a neighborhood that contains no other points of Acap A

𝐴

besides aa

𝑎

. Isolated points are part of the closure of a set, but they are never part of its derived set.

Relationship with the closure of a set

The derived set is closely related to the closure of a set, denoted Ācap A bar

𝐴̄

. The closure of a set is the union of the set itself and its derived set.Ā=A∪A′cap A bar equals cap A union cap A prime

𝐴̄=𝐴∪𝐴′

This equation provides a fundamental connection between these two topological concepts.

Proof of the relationshipTo prove Ā=A∪A′cap A bar equals cap A union cap A prime

𝐴̄=𝐴∪𝐴′

, one must show that every point in the closure is either in Acap A

𝐴

or in A′cap A prime

𝐴′

, and conversely.

• **If x∈Āx is an element of cap A bar

  𝑥∈𝐴̄** , then every open neighborhood of xx

  𝑥

  intersects Acap A

  𝐴

  . There are two possibilities:

  1. x∈Ax is an element of cap A

     𝑥∈𝐴

     . In this case, x∈A∪A′x is an element of cap A union cap A prime

     𝑥∈𝐴∪𝐴′

     .

  2. x∉Ax is not an element of cap A

     𝑥∉𝐴

     . Since every open neighborhood of xx

     𝑥

     intersects Acap A

     𝐴

     , and xx

     𝑥

     is not in Acap A

     𝐴

     , it must be that every open neighborhood of xx

     𝑥

     intersects A∖{x}cap A ∖ the set x end-set

     𝐴∖{𝑥}

     . This is the definition of a limit point, so x∈A′x is an element of cap A prime

     𝑥∈𝐴′

     . In either case, x∈A∪A′x is an element of cap A union cap A prime

     𝑥∈𝐴∪𝐴′

     .

• **If x∈A∪A′x is an element of cap A union cap A prime

  𝑥∈𝐴∪𝐴′** , then either x∈Ax is an element of cap A

  𝑥∈𝐴

  or x∈A′x is an element of cap A prime

  𝑥∈𝐴′

  .

  1. If x∈Ax is an element of cap A

     𝑥∈𝐴

     , then xx

     𝑥

     is in the closure by definition.

  2. If x∈A′x is an element of cap A prime

     𝑥∈𝐴′

     , then every open neighborhood of xx

     𝑥

     intersects A∖{x}cap A ∖ the set x end-set

     𝐴∖{𝑥}

     , and therefore also intersects Acap A

     𝐴

     . This means xx

     𝑥

     is in the closure.In both cases, x∈Āx is an element of cap A bar

     𝑥∈𝐴̄

     .

Key properties of derived sets

The derived set operator has several properties that make it a powerful tool in topology:

• Monotonicity: If A⊆Bcap A is a subset of or equal to cap B

  𝐴⊆𝐵

  , then A′⊆B′cap A prime is a subset of or equal to cap B prime

  𝐴′⊆𝐵′

  .

• Union: The derived set of the union of two sets is the union of their derived sets: (A∪B)′=A′∪B′open paren cap A union cap B close paren prime equals cap A prime union cap B prime

  (𝐴∪𝐵)′=𝐴′∪𝐵′

  .

• Finite sets: The derived set of any finite, non-empty set is empty. This is because every point in a finite set can be isolated with an open neighborhood that contains no other points of the set.

• **Closure under the operator:**A′∪A′′cap A prime union cap A double prime

  𝐴′∪𝐴′′

  is not always equal to A′cap A prime

  𝐴′

  , but for T1cap T sub 1

  𝑇1

  spaces, the derived set is always a closed set, so A′′⊆A′cap A double prime is a subset of or equal to cap A prime

  𝐴′′⊆𝐴′

  .

Examples of derived sets

To understand the derived set, it's helpful to consider several examples in different topological spaces.

Example 1: The real numbers with the standard topology

• **Set:**A=(0,1)∪{2}cap A equals open paren 0 comma 1 close paren union the set 2 end-set

  𝐴=(0,1)∪{2}

  • Limit points: Every point in the closed interval [0,1]open bracket 0 comma 1 close bracket

    [0,1]

    is a limit point. For any x∈[0,1]x is an element of open bracket 0 comma 1 close bracket

    𝑥∈[0,1]

    , any open neighborhood will contain points from (0,1)open paren 0 comma 1 close paren

    (0,1)

    other than xx

    𝑥

    .

  • **Derived set:**A′=[0,1]cap A prime equals open bracket 0 comma 1 close bracket

    𝐴′=[0,1]

    .

  • **Closure:**Ā=A∪A′=(0,1)∪{2}∪[0,1]=[0,1]∪{2}cap A bar equals cap A union cap A prime equals open paren 0 comma 1 close paren union the set 2 end-set union open bracket 0 comma 1 close bracket equals open bracket 0 comma 1 close bracket union the set 2 end-set

    𝐴̄=𝐴∪𝐴′=(0,1)∪{2}∪[0,1]=[0,1]∪{2}

    .

• **Set:**B={1n∣n∈Z+}cap B equals the set of all 1 over n end-fraction such that n is an element of the positive integers end-set

  𝐵={1𝑛∣𝑛∈ℤ+}

  • Limit points: The only point that can be "approached" by elements of this set is 0. No open interval around 0 can avoid containing some element of Bcap B

    𝐵

    . However, for any other point xx

    𝑥

    , an open neighborhood can be found that excludes all but possibly one element of Bcap B

    𝐵

    .

  • **Derived set:**B′={0}cap B prime equals the set 0 end-set

    𝐵′={0}

    .

  • **Closure:**B̄=B∪B′={1n}∪{0}cap B bar equals cap B union cap B prime equals the set 1 over n end-fraction end-set union the set 0 end-set

    𝐵̄=𝐵∪𝐵′={1𝑛}∪{0}

    .

Example 2: A finite set with a non-trivial topologyLet X={a,b,c}cap X equals the set a comma b comma c end-set

𝑋={𝑎,𝑏,𝑐}

with the open sets τ={∅,{a},{c},{a,c},X}tau equals the set the empty set comma the set a end-set comma the set c end-set comma the set a comma c end-set comma cap X end-set

𝜏={∅,{𝑎},{𝑐},{𝑎,𝑐},𝑋}

. Let A={a}cap A equals the set a end-set

𝐴={𝑎}

.

• Limit points:

  • For point aa

    𝑎

    : Every open neighborhood of aa

    𝑎

    is either {a}the set a end-set

    {𝑎}

    or {a,c}the set a comma c end-set

    {𝑎,𝑐}

    or Xcap X

    𝑋

    . The only one that contains other points is Xcap X

    𝑋

    . The intersection of Xcap X

    𝑋

    and A∖{a}cap A ∖ the set a end-set

    𝐴∖{𝑎}

    is empty. Thus aa

    𝑎

    is not a limit point of Acap A

    𝐴

    .

  • For point bb

    𝑏

    : The open neighborhoods of bb

    𝑏

    are {b,c}the set b comma c end-set

    {𝑏,𝑐}

    and Xcap X

    𝑋

    . The intersection of {b,c}the set b comma c end-set

    {𝑏,𝑐}

    and A∖{b}cap A ∖ the set b end-set

    𝐴∖{𝑏}

    is empty. Thus bb

    𝑏

    is not a limit point of Acap A

    𝐴

    .

  • For point cc

    𝑐

    : The open neighborhoods of cc

    𝑐

    are {c}the set c end-set

    {𝑐}

    , {a,c}the set a comma c end-set

    {𝑎,𝑐}

    , and Xcap X

    𝑋

    . The intersection of {c}the set c end-set

    {𝑐}

    and A∖{c}cap A ∖ the set c end-set

    𝐴∖{𝑐}

    is empty. Thus cc

    𝑐

    is not a limit point of Acap A

    𝐴

    .

• **Derived set:**A′=∅cap A prime equals the empty set

  𝐴′=∅

  .

Historical context

The concept of the derived set was first introduced by Georg Cantor in 1872 during his development of set theory. Cantor was deeply interested in the properties of subsets of the real line, and the derived set was a key tool he used to study the accumulation of points. He developed a remarkable theory of transfinite ordinals by repeatedly taking the derived set of a set, a process known as Cantor's construction. This work laid foundational stones for modern point-set topology.