Compactness and connectedness: Linear Functional Analysis - 2 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

3.2 CONNECTEDNESS IN CLOSURE SPACES.

In this section we introduce and study connectedness.

Definition 3.2.1

(X,c) is said to be disconnected if it can be written as two disjoint nonempty subsets A and B such that cA U cB =X, cA ∩ cB = φ and cA and cB are nonempty. A space which is not disconnected is said to be connected.

Example 3.2.2

X = {a,b,c}

c can be defined on X such that

c {a}={a,b}, c{b}=c{c}=c{b,c}={b,c}, c{a,b}=c{a,c}=cX=X, cφ=φ)

Then c is a closure operation on X.

Here (X,c) is connected because we can not find nonempty subsets A and B such that cA U cB=X and cA ∩ cB=φ.

Definition 3.2.3

(X,c) is said to be feebly disconnected if it can be written as two disjoint nonempty subsets A and B such that A U cB = cA U B=X and cA ∩ B=φ=A ∩ cB.

Note 3.2.4

It is clear that (X,c) is disconnected implies (X,c) is feebly disconnected. The following example shows that the converse is not true

Example 3.2.5

X={a,b,c}

c{a}={a,c},c{b}=c{c}=c{b,c}={b,c},c{a,b}=c{a,c}=cX=X,cφ=φ

c is a closure operation on X.

Here (X,c) is feebly disconnected, but not disconnected.

Result 3.2.6

(x,t) is disconnected ⇒ (X,c) is disconnected.

Proof

(X,t) is disconnected implies that it is the union of two disjoint nonempty subsets A and B such that clA U clB =X, clA ∩ clB=φ and clA, clB are nonempty. clA∩clB=φ)). So cA ∩ cB=φ. That is (X,c) is disconnected.

Note 3.2.7

(X,t) is connected need not imply that (X,c) is connected.

Example

X={a,b,c} . Let c be a closure operation defined on X in such a way that

c {a} = {a},c {b}= {b,c},c {c}= c {a .b} = c {b ,c }= cX = X , cφ=φ)

(X,t)={X,φ,{b,c}}

Here (X,c) is disconnected, but (X,t) is connected.

Remark

Connectedness of a subspace Y of (X,c) can be defined in the same manner.

Note 3.2.8

Let (X ,c) be a closure space and Y be a connected subset of (X ,c). Then cY  need not be connected.

Example 3.2.9

X={a,b,c,d,e}

Let c be defined on X such that

c{a}= {a},c{b}= {a,b,c},c{c}= {b,c},c{d} = {b,c,d },

c{a,b}=c{a,c}=c{b,c}=c{a,b,c}={a,b,c},

c{c,d}={b,c,d}, c{a,d}=c{b,d}=c{a,b,d}=c{a,c,d}=c{b,c,d}=c{a,b,c,d}={a,b,c,d},

c {e}= c {a,e}= c {b,e}= c {c ,e}= c{d,e}= c {a ,b ,e }= c {a ,c ,e }= c {a ,d ,e }= c {b ,c ,e }

= c {c,d ,e} =c{b,d,e}=c{a,b,d,e}=c{a,c,d,e}=c{b,c,d,e}=cX=X,cφ=φ

Here Y={b,c} is connected.

cY={a,b,c}; if c' isthe induced closure operation on cY, then 

c'{a}={a},c'{c}={b,c},c'{b}:=;c'{a,b}=c'{b,c}=c'{a,c}=c'cY=cY. 

cY is disconnected.

Note 3.2.10

If cA and cB form a separation o f X and i f Y is a connected subset o f X, then Y need not be entirely within either cA or cB.

Example 3.2.11

X={a,b,c}

Let c be a closure operation defined on X such that 

c{a}={a},c{b}={b,c} c{c}={a,c},c{a,b}=c{b,c}=cX=X,c{a,c}={a,c}.

Y={a,c} is connected •

Note 3.2.12

The image of a connected space under a c -c' morphism need not be connected. 

Example

Let X={a,b,c,d,e}. A closure operation c is defined on X as in Example 3.2.9

Let Y={a,b,c}

c' be defined on Y such that

c'{a}={a},c'{b}={b,c},c'{c}=c'{a,b}=c'{b,c}=c'{a,c}=c'X=X c'φ =φ

Let f be a map from (X,c) into (Y,c') defined in such a way that f(a)=a, f(b)=c, f(c)-b, 

f(d)=c, f(e)=c.

Here f is a c-c' moiphism. But f(X) is disconnected.

Result 3.2.13 

Suppose c, is a closure operator on Y with degree k and f is a c-c, morphism from (X ,c) to (Y,c₁). If c₁ ^k(A) and c₁^k(B) form a separation of Y, then c (f^-1(c₁^k(A)) and c(f^-1(c₁^k(B)) form a separation on X.

Proof

In similar manner

 form a separation on X •

Result 3.2.14

Let (X,c) be connected and f is a c-c₁ morphism from (X,c) on to (Y, c₁). Then(Y,t₁) is connected.

Proof

 morphism and we get f is c - cl₁ morphism. Suppose cl₁ A and cl₁ B form a separation on Y. Then   and   By th e above  result   form a separation on X. This is a contradiction . Hence (Y,t) is connected.

3.3 PATHWISE AND LOCAL CONNECTEDNESS

In this section we define and study pathwise connectedness and local connnectedness.

Definition 3.3.1

A space (X,c) is pathwise connected if and only if for any two points x and y in X, there is a cl₁ -cmorphism f : I → X such that f(o) = x and f (l) = y where cl₁ is th e usual closure on I, f is called a path from x to y.

Result 3.3.2

(X,c) is pathwise connected implies (X,t) is pathwise connected.

Proof

If (X,c) is pathwise connected, then for any two points x and y in X there is a cl₁ - cmorphism f : I → X such th a t f(0)=x and f(l) = y. If f is cl₁-cmorphism ,then f is cl₁-cl morphism. Therefore (X,t) is pathwise connected.

Note 3.3.3

The converse of the above result is not true.

Note 3.3.4

Pathwise connected space need not be a connected space.

Definition 3.3.5

A space X is said to be locally connected at x if for every neighbourhood U of x, there is a connected neighbourhood V of x contained in U. If X is locally connected at each of its points, then X is said to be locally connected.

Definintion 3.3.6

A space X is said to be locally path connected at x if for every neighbourhood U of x, there is a path connected neighbourhood V of x contained in U. If X is locally path connected at each of its points, then it is said to be locally path connected.

Note 3.3.7

A space (X,c) is locally connected need not imply that (X,t) is locally connected and vice-versa.

A parallel study of the above concepts in the set up of closure spaces is interesting; however we are not attempting it in this thesis.

3.4. COMPACTNESS AND CONNECTEDNESS IN MONOTONE SPACES

Definition 3.4.1

Let (X,c_*) be a monotone space. F be a proper filter on X and x be an element of X. We shall say that x is a cluster point of F in (X,c_*) if x belongs to ∩{c_*F : F∈F}. That is each neighbourhood of x intersects each F ∈ F.

Definition 3.4.2

A monotone space (X,c_*) is said to be compact, if every proper filter of sets on X has a cluster point in X.

Remark 3.4.3

It is clear that if (X,c_*) is compact, then (X,c) is compact but the converse is not true.

Result 3.4.4

Any image under a c-c_* morphism of a compact monotone space (X,c_*) onto a monotone space (Y,c_*) is compact.

The proof is similar to the Proof of 41 A - 15 in [CE₂] .

Result 3.4.5

Every closed subspace of a compact monotone space is compact.

The proof is similar to the Proof of 41 A - 10 in [CE₂] .

Result 3.4.6

If (Y , c') is a compact subspace of a Hausdorff monotone space (X ,c_*),then Y is closed in X.

The proof is similar to the Proof of 41 A-11 in [CE₂] .

Definition 3.4.7

A monotone space (X,c_*) is said to be discomiected if it can be written as two disjoint nonempty subsets A and B such that c_*A U c_*B = X, c_*A ∩ c_*B=φ). A space which is not disconnected is said to be connected.

Remark 3.4.8

(X,c) is disconnected implies (X,c*₎ is disconnected,and the converse is not true. 

Example 3.4. 9

X={a,b,c}

c_* be defined on X such that

c_*{a}={a},c_*{b}={b,c}, c_*{c}={b,c}, c_*{a,b}=c_*{b,c}=c{a,c}=c_*X=X,c_*φ=φ

c_* is a monotone operator.

(X,c_*) is disconnected. But (X,c) is connected.