Compactness and connectedness: Linear Functional Analysis - 1 | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

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INTRODCUTION

In this chapter we firstly describe the fundamental properties of compactness. We define compact closure spaces and study some properties of compactness.

Cech defined closure space X to be compact if the intersection of the closures of sets belonging to any proper filter in X is nonempty. He proved some properties of compactness in closure spaces [CE₂], In section 1 of this chapter, we find the relat ion between compactness in (X,c) and (X,t) and prove some related results.

Cech described the concept of connectedness in [CE₂] as "a subset A of a closure space X is said to be connected in X if A is not the union of two nonempty semi-separated subsets of X. That is A = A₁ U A₂ ,(cA₁∩ A₂) U (A₁ ∩ cA₂)= φ implies that A₁ = φ) or A₂ = φ" . It can be easily seen that this is precisely the connectedness of the associated topological space. Plastria, F obtained certain conditions which imply the connectedness of simple extensions [P]; it has been proved that local connectedness of certain subspaces implies the local connectedness of simple extensions.

We define the concept of connectedness in section 3.2 in a slightly different and perhaps more reasonable way and prove some results in connectedness. We note that the image of a connected space under a c -c₁ morphism need not be connected.

In section 3.3 we introduce the concepts of local connectedness and path connectedness. We also define compactness and connectedness in monotone spaces in 
section 3.4.

3.1SOME PROPERTIES OF COMPACTNESS

The following definitions and results are due to E.Cech.

Definitions 3.1.1

(i) Let (X ,c) be a closure 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 ∩ (cF : F ∈ F) , that is if each neighbourhood of x in tersects each F ∈ F.

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

Results 3.1.2

(i) For a closure space (X,c) to be compact, it is necessary and sufficient that every interior cover V of (X,c) has a finite subcover.

(ii) Any image under a c-morphism of a compact space (X,c) is compact.

(iii) If (Y,c) is a compact subspace of a Hausdorff closure space (X,c), then Y is closed in (X,c).

(iv) Every closed subspace of a compact space (X,c) is compact.

Result 3.1.3

If (X,c) is compact, then (X,t) is compact.

Proof

Let (X,c) be compact. Then every proper filter of sets on X has a cluster point in X. Let F be a proper filter of sets on X and x be a cluster point. Then x ∈ ∩ (cF), Fe F. That is ∩ (cF) ≠ φ but cF ⊂ clF for every F ∈ F. Then ∩ (clF) ≠ φ . So ( X ,t) is compact.

Note 3.1.4

The converse of the above result is not true.

Example

Consider X = N x N U {x,y} U {a;: i ∈ N} U {b_j : j ∈ N}, 

a_i's, b_j's, x,y are all distinct and do not belong to N x N.

Let c be defined on X as in Example 2.1.11 

Let A_k = {(m, m) : m > k} for k∈N .

The family F = {A_k : k ∈ N} is a filter base. 

cA_k = A_k, for every A_k ∈ F but ∩^∞_k=1 cA_k = φ.

So (X,c) is not compact. But (X,t) is compact as can be proved easily.

Result 3.1.5

Any image under a c -c' morphism of a compact closure space (X , c) is compact  in the associated topology of c'.

Using the Result 3.1.2 (ii) and the Result 3.1.3, we get this result.

Note 3.1.6

If (X , cl) is compact and f : (X , cl) → (Y , c') is a surjective c- c' morphism, then  (Y,c') need not necessarily be compact.

Result 3.1.7

The associated space (Y, t') o f a compact closure space (Y, c') is closed as a subspace o f the H au sdorff space (X, c)

Using the Result 3.1.2 (iii) and cA=X ⇒clA=X, we get the above result.

Result 3.1.8

Every closed closure subspace of an associated topological space (X, t) of a compact closure space (X , c) is compact.

Proof

Let (Y, c') be a closed subspace of a compact space (X, t) . Let F be a proper filter on (Y,c') . Let us consider the smallest filter G on X containing F. F is a filter base for G. Since cl Y = Y, we have c'A = clA for each A ⊂ Y and hence ∩ (c'F) = ∩ (cl F). Therefore ∩ (cl F) = ∩ (cl G). Since (X, t) is compact ∩ (cl G)≠φ. That is ∩(c' F)≠φ.

Corollary 3.1.9

Closed subspace (Y,t') of compact space (X,c) is compact.

Result 3.1.10

(X,c) is compact. Y C X. Then cY is compact.

Proof

Let c' be the closure on cY induced by c. Let F be a filter on cY. We have to prove that ∩(c'F), F ∈ F is nonempty. {cF ∩ cY : FeF) is a filter b ase on X. Since X is compact, ∩(cF ∩ cY) is nonempty. So ∩ c'F = ∩(cF ∩ cY)≠φ.

Definition 3.1.11

A closure space (X,c) is locally compact if and only if each point in X has a neighbourhood base consisting of compact sets.

Note 3.1.12

(X,c) is locally compact does not imply that (X,t) is locally compact and vice-versa.

Result 3.1.13

Let (X,c) be locally comapct. If f is an open c-c' morphism from (X,c) onto (Y,c'), then Y is locally compact.

Proof

Suppose y ∈ Y . Let V be a n eighbourhood of y. Take x ∈ f^-1(y). Since f is c-c' morphism and X is locally compact, we can find a compact neighbourhood U such that f(U)⊂ V . x ∈ Int_x U so y ∈ f(Int_x U) ⊂ f(U) . Since f is open, f(Int_x U) is a neighbourhood of g. Hence 1(U) is a compact neighbourhood of y contained in V.