The Use of Filters in Topology

Home , Base (topology), Closure (topology), Filters in topology, Filter (mathematics), Net (mathematics), Partially ordered set

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Electronic Theses and Dissertations, 2004-2019

2004

The Use Of Filters In Topology

Abdellatif Dasser University of Central Florida

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The Use of Filters in Topology

ABDELLATIF DASSER B.S. University of Central Florida, 2002

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the Department of Mathematics in the College of Arts and Sciences at the University of Central Florida Orlando, Florida

Fall Term 2004 ABSTRACT

Sequences are sufficient to describe topological properties in metric spaces or, more generally, topological spaces having a countable base for the topology. However, filters or nets are needed in more abstract spaces. Nets are more natural extension of sequences but are generally less friendly to work with since quite often two nets have distinct directed sets for domains. Operations involving filters are set theoretic and generally certain to filters on the same set. The concept of a filter was introduced by H.

Cartan in 1937 and an excellent treatment of the subject can be found in N. Bourbaki

(1940).

ii ACKNOWLEDGEMENTS

I would like to express my gratitude to my supervisor, Dr. Richardson, for many insightful conversations during the development of the ideas in this thesis, his understanding, endless patience, and encouragement.

I would also like to think my defense committee members, Dr. Mohapatra and

Dr. Han for asking me excellent questions .

Finally, but not least, I would like to think Ahmed Ameur for the friendship and comic relief when things started to get difficult.

iii TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION AND EXAMPLES ...... 1 CHAPTER 2 FILTERS...... 4 CHAPTER 3 ULTRAFILTERS ...... 8 CHAPTER 4 CONVERGENCE AND FILTERS ...... 12 CHAPTER 5 COMPACTNESS AND FILTERS...... 15 CHAPTER 6 INITIAL STRUCTURES ...... 17 CONCLUSION...... 19 LIST OF REFERENCES...... 20

iv CHAPTER 1

INTRODUCTION AND EXAMPLES

The study of filters is a very natural way to describe convergence in general

topological space. Filters were introduced in 1937 by Cartan (1937 a,b). Bourbaki (1940)

employed filters in order to prove several results in their text . In the same year Tukey

(1940) studied sets, filters, and various modifications of the two concepts. A complete

reliance on filters for the development of topology can be found in Kowalsky (1961).

There are traces of the concept of filters as early as 1914 in Root’s article. More recently,

filters play a fundamental role in the development of fuzzy spaces which have

applications in computer science and engineering. Filters are also an important tool used

by researchers describing non-topological convergence notions in functional analysis.

(e.g see Beattie and Butzmann,2002). Moreover, Preuss (2002) has applied filters

throughout his book on categorical topology.

The purpose of this paper is to provide thorough discussion of filters and their

applications. Filters are used in general topology to characterize such important concepts

as continuity, initial and final structures, compactness, etc.

The following examples are given to show that sequences are not sufficient to

characterize points of closure, continuity, and compactness.

Example 1.1 let X to be an uncountable set and fix x0 ∈ X .

c Define τ = {A ⊆ X : ()x0 ∉ A or (x0 ∈ Aand A is countable)}. Then τ is a topology for X.

(a) φ , X ∈τ

1 (b) A, B ∈τ implies that A ∩ B ∈τ

c ⎛ ⎞ then ⎜ A ⎟ = Ac ⊆ Ac which is countable. Thus, A ∈τ . ∪ α ∩ α α0 ∪ α ⎝ α ⎠ α α

τ We claim that xn ⎯⎯→ x if and only if xn = x eventually ( ie xn = x ∀ n ≥ N ).

τ (a) Suppose that x ≠ x0 . Since{}x ∈τ , xn ⎯⎯→ x if and only if xn = x eventually.

(b) Suppose that x = x0 and xn ≠ x0 for infinitely many n. Define F={xn : xn ≠ x0 }. Then

c c F ∈τ and xn ∈ F eventually fails to hold. Hence xn ≠ x0 infinitely often. Therefore,

xn does not converge to x0 . Conversely, if xn does not converge to x0 , then there exist

O ∈τ , x0 ∈ O such that xn ∉ O infinitely often. That is, xn ≠ x0 infinitely often. Hence,

τ xn ⎯⎯→ x0 if and only if xn = x0 eventually.

c c τ A: x0 ∈{x0 } but it does not exist a sequence {xn } in {x0 } such that xn ⎯⎯→ x0 .

c If xn ∈{x0 }, then xn ≠ x0 for all n ≥ 1 and by above results, xn does not converge to

c x0 . Hence, there is no sequence contained in {x0 } that converges to x0 . However,

c c x0 ∈{x0 } since O ∩{}x0 ≠ φ for each O ∈τ , x0 ∈ O . And therefore, sequences do not characterize points of closure.

B: Let σ be the discrete topology for X, ie σ is the set of all subsets of X. One can see

σ that xn ⎯⎯→ x0 if and only if xn = x0 eventually. Hence, σ and τ have the same convergent sequences. Let the Id: (X ,τ ) → (X ,σ ) denote the identity function. The

2 function Id is sequentially continuous since τ and σ have the same convergent

sequences. However, since τ ⊂ σ , the above function is not continuous. Hence,

sequences do not characterize continuity.

Example1.2 Let (X ,d )be a metric space that is not compact. And let (X * ,τ * ) be the

Stone-Cech compactification of (X ,d) . Since (X ,d ) is not sequentially compact, there

exist {xn } contained in X which has no convergent subsequence in (X ,d) . It is known that no sequence contained in X converges to a point in X * − X . Hence, (X * ,τ * ) is not sequentially compact. Therefore, sequences do not characterize compactness.

3 CHAPTER 2

FILTERS

Definition 2.1 Consider an arbitrary set X. A set τ of subsets of X satisfying the

conditions:

(a) φ ∈τ and X ∈τ

(b) U ∩V ∈τ whenever u ∈τ and V ∈τ

(c) The union of the members of an arbitrary subset of τ belongs to τ is called a topology on X. A topological space is a pair (X ,τ ) where τ is a topology on X.

The members of τ are called open sets.

Definition 2.2 Consider a set X ≠ φ. A filter F on X is a set of subsets of X satisfying the conditions:

(a) F ≠ φ and φ ∉F

(b) If A, B ∈F then A ∩ B ∈F

A subset β ⊆ F is called a base for the filter F if every member of F contains some member of β. The definition of a filter base for some filter is as follows:

Definition 2.3: β is called a base for a filter on X if and only if β is a set of subsets of

X satisfying the conditions:

(a) β ≠ φ , φ ∉ β

4 (b) B1 , B2∈⇒β ∃B3∈β such that B3 ⊆ B1 ∩ B2 .

Example 2.4 Let X ≠ φ be an arbitrary set. Fix x0 ∈ X and then

x& 0 = {}A :A ⊆ X and x0 ∈ A is a filter on X.

Note that {}x0 ∈ x& 0 .

Example 2.5 Fix a set φ ⊄ A0 ⊆ X , then A& 0 ={B ⊆ X : B ⊇ A0 } is a filter on X. In

particular, if A0 = X, X& ={X} is the “smallest” possible filter on X.

Example 2.6 If X is any nonempty set and {xn} is a sequence in X. Define

Bn = {xk : k ≥ n ≥ 1}. Then, F = {A ⊆ X : A ⊇ Bn ∃n ≥ 1} is a filter on X and is called the

elementary filter determined by {xn}.

Example 2.7 If X is an infinite set then F={F ⊆ X : F cis finite} a filter on X and is

called the cofinite or Fréchet filter.

Example 2.8 If X is a topological space and x ∈ X , then the family U(x) of all

neighborhoods of x is a filter and is called the neighborhood filter of x.

Example 2.9 The family of all ‘tails’ of the sequence {xn} on X is a base for the

corresponding elementary filter; a tail is the set of the form BN = {xn : n≥ N} .

Example 2.10 The family {{}x }is a base for the filter x& on X.

Let F (X) denote the set of all filters on a set X and F, G ∈ F (X). We call a filter G

finer than the filter F if F ⊆ G, we also call F coarser that G. Note that F = {X} is the

coarsest member in F (X). It is easy to verify that ( F(X ),≤ ) is a poset.

5 Also, F( X ,≤ ) is not linearly ordered since x& ⊄ y& or y& ⊄ x& . Before discussing filter

and convergence, one wants to prove and define various things about filters. DeMorgan’s

law states that if X ≠ φ and{Aα :α ∈ I } is a collection of subsets of X. Then:

c c (a) ()Ai = ∩ Aα ∪ α∈I α∈I

c c (b) ()Ai = ∪ Aα ∩ α∈I α∈I

Proposition 2.11 Assume that X ≠ φ andFα ∈ F(X ) , α ∈ I (index set) then

∩Fα ∈ F(X ) is the finest filter on X which is coarser than each Fα , α ∈ I .

Proof (a): Note that φ ∉F , α ∈ I , implies that F and also X belongs to α φ ∉ ∩ α α∈I

eachFα , α ∈ I .

Thus X∈ F and it follows that F ≠ φ . ∩ α ∩ α ∂∈I ∂∈I

(b): Let A, B ∈ F ; then A and B belong to each F . Therefore, A ∩ B belongs to ∩ α α α∈I each F ,α ∈ I , and thus A ∩ B∈ F . α ∩ α α∈I

(c): Let A∈ F ; then A ∈ eachF , α ∈ I . Let B ⊇ A thus B belongs to each F ∩ α α α α∈I

since B is an over set of A. It follows that B∈ F . Since (a), (b) and (c) are satisfied, ∩ α α∈I

F is a filter. Clearly F ⊆ F , for each α ∈I. ∩ α ∩ α α α∈I α∈I

6 Next, letG ⊆ F for each α ∈ I and let us prove that ⊆ F . Let A ∈ ; thus A α G ∩ α G α∈I belongs to eachF , α ∈ I , and then A∈ F . It follows that ⊆ F .▫ α ∩ α G ∩ α α∈I α∈I

In general, the union of two filters may or may not be a filter. For example, if F and G

may contain disjoint members.

Proposition 2.12 LetFα , α ∈ I , be filters on X. Then

F = A ⊆ X : A = F for some F ∈F ∩ α {}∪ α α α α∈I α∈I

Proof Let B∈ F ; then B belongs to eachF and thus B = F by choosing each ∩ α α ∪ α α∈I α∈I

F = B . Conversely, Let B∈ A ⊆ X : A = F for some F ∈F , α {}∪ α α α α∈I thus B = F for some F ∈ F and thus B belongs to each F . Hence ∪ α α α α α∈I

B ∈ F ▫ ∩ α α∈I

Letφ ⊆ X ⊆ Y . If F ∈ F(X ) , then F is a filter base on Y. That is

{A ⊆ Y : F ⊆ A∃F ∈F } is a filter on Y generated by F. The generated filter is denoted by

[F ]. Conversely, if G is a filter on Y and GX∩ ≠ φ for each G∈ G , then

F = {G ∩ X : G ∈G}∈ F(X). This filter is called the induced filter on X, or the trace of G on X.

Example 2.13 Let X =[0,1],Y=R and let G be the filter on Y whose base is

{()−εε,:ε>0}. Then the trace of G on X is the filter on X having a base {}[]0,ε : ε > 0 .

7 CHAPTER 3

ULTRAFILTERS

Definition 3.1 An ultrafilter is a maximal filter in the poset (FX( ),≤) ,where the

ordering F ≤ G means that F ⊆ G. That is a filter U on X is an ultrafilter provided

U ⊆ G implies that U =G.

Proposition 3.2 ( Zorn’s lemma ) If X is partially ordered set in which every linearly

ordered subset ( any two elements are comparable ) has an upper bound , then X has a

maximal element. That is, there exists x∈ Xsuch that there is no y≠ x with x ≤ y .

Proposition 3.3 Let X be a set and F a filter on X. Then there exists an ultrafilter U on X

that is finer than F..

Proof: Consider the family P={}G∈ F(X ) :Gis a filter that finerthan F . The family P

is partially ordered by⊆ . Suppose that C = {Gα :α ∈ I}is a chain in P. That is C is

linearly ordered subset of P for each G∈ P .

Denote H= ∪ {}Gα :α ∈ I ={}A : A∈Gα ∃α ∈ I

(a) F ∈P by construction, thus H ≠ φ

(b) φ ∉H since φ ∉G, for eachG∈ P

(c ) Let A, B∈H; then A∈ Gβ , B ∈ Gβ for some α, β . Now, eitherGβ ⊆ Gα or Gβ ⊆ Gα

holds. Then A, B ∈ Gβ and thus A ∩ B∈ Gβ sinceGβ is a filter. Then, A ∩ B∈H.

(d) If A∈H then A∈ Gα for some Gα ∈ C . If B ⊇ A then B∈ Gα and thus B∈ H.

8 Therefore, H is a filter and hence an upper bound for C in P The partially ordered set

P satisfies the assumptions of Zorn’s lemma; hence there is a maximal element

U∈P. Therefore, U is an ultrafilter containing F.▫

Proposition 3.4 Let F be a filter on a set X; then, the following are equivalent:

(a) F is an ultrafilter.

(b) For any two subsets A and B of X we have: If A U B ∈F then A∈F or B∈F . c) For every subset A of X either A∈F or A c ∈F .

Proof (a) ⇒ (b): Assume A U B∈F and A∉F and B∉F.

Define G = {C ⊆ X : A ∪ C ∈F }; then G ∈ F(X) .Further, F ⊆ G , B∈ G and thus F ≠ G. But F is an ultrafilter. Thus, there is a contradiction. Therefore, A∈F or B

∈F .

(b) ⇒ (c): Clearly A ∪ Ac = X ∈F ; therefore A ∈F or Ac ∈F using(b).

Ac ∉F because A has nonempty intersection with every element of F. It follows that

A∈F . Thus, G = F. ▫

Proposition 3.5 Let F ∈F(X). Then:

F = ∩ {}U : U ⊇ F ,U is anultrafilter on X

Proof Clearly F ⊆ ∩ {}U : U ⊇ F ,U is anultrafilter on X . Assume that there exists an

A∈∩{}U : U ⊇ F ,U is anultrafilter on X such that A∉F. Then F ∩ A c ≠ φ for each F∈F and thus β = {F ∩ Ac : F ∈F } is a base for a filter G. Note that F ⊆ G.

9 c Let UG be an ultrafilter containing G. Since, F ⊆ UG , A∈UG . However, A ∈G

c ⊆UG and thus A ∩ A∈UG , a contradiction. Therefore,

F = ∩ {}U : U ⊇ F ,U is anultrafilter on X .

Proposition 3.6 Let U be an ultrafilter on the set X. If AA12..., An are subsets of X such

n that A belongs to , then at least one of the sets A belongs to ∪ i U i U. i=1

c Proof If no Ai belongs to U, then Ai belongs to U for i = 1,2,...., n by Proposition 2.3,

n and hence ∩ Ac = (∪A )c belongs to , which is impossible since A is given to i i U ∪ i i=1

belong to U..

Proposition 3.7 let U be an ultrafilter on X and A ⊆ X such that

U ∩ A ≠ φ for all U ∈U. Then A ∈U.

Proof Assume that A ∉U and define β = { U ∩ A :U ∈U }. Note that

φ ∉ β and β ≠ φ since U ∩ A ≠ φ for all U∈U . Also,

((U1 ∩ A) ∩ U 2 ∩ A)= ()U1 ∩U 2 ∩ A∈ β since ( U1 ∩U 2 )∈U. Thus β is a base

filter for some filter G.Note that A∈G because A ⊇ U ∩ A and A∈ U.

Therefore, U ⊂ G, a contradiction and hence A∈U.

Proposition 3.8 Let f : X → Y be a function and U is an ultrafilter on X. Then f (U ) is

an ultrafilter on Y.

Proof Assume that Af∉ (U ); then for each U∈U, Ac ∩ f (U ) ≠ φ .

10 Hence, f −1 (Ac )∩U ≠ φ for each U∈U and thus by Proposition 2.6, f −1 (Ac )∈U.

Therefore, f ( f −1 (Ac ))∈ f (U ) and thus Ac ∈ f (U ) since f ( f −1 (Ac )) ⊆ Ac . Hence, f (U) is an ultrafilter on Y.▫

11 CHAPTER 4

CONVERGENCE AND FILTERS

Definition 4.1 Let (X ,τ ) be a topological space and let U (x) denote the neighborhood filter at x. A filter F on X converges to x if U (x) ⊆ F.

Proposition 4.2 Let A be a subset of a topological space X. Then, for x∈ X, x∈ Ā if and only if there exists a filter on X which contains A and converges to x.

Proof Assume that x∈ Ā. Then any neighborhood of x has a nonempty intersection with

A. Now all the sets A ∩ U, where U is a neighborhood of x, form a filter base, and the

corresponding filter converges to x. Conversely, assume that F is a filter containing A and converging to x . Choose any neighborhood U of x. Then U∈F , and thus

U ∩ A ≠ φ since A ∈F. This proves that x∈ Ā.

Definition 4.3 A topological space ( X ,τ ) is Hausdorff or T 2 provided if x ≠ y , then

there exist sets Ox ,Oy ∈τ such that x ∈ Ox , y ∈ Oy and Ox ∩ Oy = φ .

Proposition 4.4 (X ,τ ) is T 2 if and only if each filter converges to at most one point, i.e.

F ⎯⎯→τ x, y implies x=y.

Proof Suppose that (X ,τ ) is Hausdorff and suppose F ⎯⎯→τ x, y where x ≠ y .

Then there exist Ox ,Oy ∈τ such that x ∈ Ox , y ∈ Oy and Ox ∩ Oy = φ .However,

τ F ⎯⎯→ x, y implies thatOx ∈F, Oy ∈F, and

12 Ox ∩ Oy =φ ∈F which is a contradiction. Therefore, each filter converges to at most one

point. Conversely, suppose that x ≠ y and assume that Ox ∩ Oy ≠ φ for each

Ox ,Oy ∈τ , x ∈ Ox , y ∈ Oy

We claim that β = {Ox ∩ O y : ∀Ox ∈τ ,Oy ∈τ , x ∈ Ox , y ∈ O y } is a base for some filter

F.Observe that (Ox ∩ O y ) ∩ (Gx ∩ G y ) = (Ox ∩ Gx ) ∩ (Oy ∩ G y ) ∈ β since (Ox ∩ Gx )

is an open set containing x and (Oy ∩ G y ) is an open set containing y. Thus β is a base

for some filter F since Ox ⊇ (Ox ∩ Oy ) implies that each Ox ∈F,

F converges to x. Likewise, Oy ⊇ (Ox ∩ Oy ) implies that Oy ∈F, and thus F converges

to y, a contradiction. Therefore, there doesn’t exist anOx and Oy such thatOx ∩ O y ≠ φ

where x ∈Ox and y ∈ Oy . Hence ( X ,τ ) is Hausdorff.

The following result shows that continuity of maps between topological spaces can be characterized in terms of convergence of filters. Recall that a mapping g: X →Y between two topological spaces is continuous at x provided that for each

neighborhood V of g ( x ), there exist a neighborhood U of x such that

g(U) ⊆ V.

Proposition 4.5 Let X, Y be topological spaces with x∈ X and g: X →Y . Then g is continuous at x if and only if whenever F is a filter such that F →

x, g(F ) →g(x).

Proof Suppose g is continuous at x and F → x. Let V be a neighborhood of g(x). By continuity there is a neighborhood U of x such that g(U) ⊆ V. Since U∈ F,

13 g(U)∈g(F ). And thus V∈ g(F ). Hence g (F ) → g(x). Conversely, suppose that whenever F → x, g(F) → g (x). Then g(U(x)) → g(x) by hypothesis. Then, for each

neighborhood V of g (x), V∈ g(U(x)). Then there exists a U ∈U(x) such that

g (U) ⊆ V and thus g is continuous at x.

14 CHAPTER 5

COMPACTNESS AND FILTERS

Recall that a topological space (X ,τ ) is compact provided each open covering of

X has a finite subcovering . It is shown below that compactness can be characterized in

terms of convergence of ultrafilters.

Proposition 5.1 Let (X ,τ ) be a topological space. Then the following statements are

equivalent:

(a) (X ,τ ) is compact

(b) Each ultrafilter on X converges

Proof (a)⇒ (b): Suppose (X ,τ ) is compact and that there exist an ultrafilter F that

doesn’t converge. Then for each x∈ X, there exists Vx ∉F V and thus CV=∈{ x : xX}

n is an open cover of X. Hence V . =X. Since F is an ultrafilter, implies that ∪ xi i=1

n V c ∈F for each x ∈ X and therefore X c = V c = φ ∈F , which is a contradiction. x ∩ xi i=1

Thus, each ultrafilter on X converges.

(b)⇒ (c): Given any filter F on X; let G be an ultrafilter containing F. Then G converges

to x in (X ,τ ), for some x∈ X. Given any neighborhood V of x and A ∈F; then A ∈G and V∈G. Hence, A ∩V ∈ G and thus AV∩ ≠ φ . Therefore, x ∈ A .

15 (c)⇒ (a): Suppose (X ,τ ) is not compact. Let C ={Oα :α ∈ J} be an open cover of X

n with no finite subcover. Then, O ≠ X , for each n. Let F be the filter on X whose ∪ αi i=1

n ⎧ c ⎫ c c base ⎨ Oα : n ≥ 1,Oα ∈ C⎬ .However, ∩{A : A∈F} ⊆ ∩Oα = ( ∪ Oα ) = X = φ , ∩ i α∈J ⎩ i=1 ⎭ which is a contradiction.

Proposition 5.2 Let f : (X ,τ ) → (Y,σ ) be a continuous function and onto. If (X ,τ ) is compact then (Y ,σ ) is compact.

Proof Let U be an ultrafilter on Y , F = f −1 (U ) and by Proposition2.2 there exist an

ultrafilter on X. G ⊇ F. Then:G⎯⎯→τ x , for some x ∈ X since ( X ,τ ) is compact.

The continuity assumption of f implies that f (G ) → f(x) according to Proposition 4.6.

−1 Since G ⊇ F then f (G) ⊇ f ( F ) =f (f ( U)) ⊇ U. Also f is onto and thus

−1 f (f (U))= U. Hence, f(G ) ⊇ U and since U is an ultrafilter f ( G ) = U.

Consequently, f (G ) → f (x) implies that U → f(x). Therefore, by

Proposition 4.1, (Y ,σ ) is compact.

16 CHAPTER 6

INITIAL STRUCTURES

fα Proposition 6.1 Consider the source X ⎯⎯→(Yα ,τ α ), α ∈ J where J, is an index class.

Then:

(a) There exists a coarsest (smallest) topology τ I on X for which each

fXα :,()τ I → (Yα,τα) is continuous, α ∈ J .

(b) Each g: (Y,σ ) → (X ,τ I ) is continuous if and only if fαο g : (Y,σ ) → (Yα ,τ α ) is continuous, for eachα ∈ J .

τ I τ I (d) Given F ∈ F(X ), F ⎯⎯→ xi if and only if fα (F )⎯⎯→ f (xα )

for each α ∈ J .

Proof A subbase for τ is S ={f −1 (O ) : O } . I α α α∈τα

(a) It easily follows that τ I is the coarsest such topology such that each fα is continuous.

(b) The composition of two continuous functions in continuous. Conversely, assume that

−1 f ο g is continuous, for each α ∈J. Let fα (Oα ) ∈S then,

−1 −1 −1 −1 −1 g ( fα (Oα )) = ( fαο g) (Oα ) ∈σ and thus, g ( fα (Oα ))∈σ

Since continuity of g: (Y,σ ) → (X ,τ I ) is determined by the subbase S for τ I , it

follows that g: (Y,σ ) → (X ,τ I ) is continuous.

17 (c) Let τ X be another topology for X obeying (b). Since id: (X ,τ I ) → (X ,τ X ) is

continuous, fα : (X ,τ X ) → (Yα ,τ α ) is continuous, for each α ∈ J . Hence by (a),τ I ⊆τ X .

Moreover, consider id: (X ,τ X ) → (X ,τ I ) .

Since fα = fαο id :()X ,τ I → (Yα ,τ α ) is continuous for each α ∈ J , the hypothesis

implies that id: (X ,τ I ) → (X ,τ X )is continuous. Hence, τ X⊆τ I and thus τ XI=τ .

τ I (d) Since each fXα :,()τ I → (Yα,τα) is continuous, F⎯⎯→ x implies that .

τα τα fα (F)⎯⎯→ fα (x) . Conversely, assume that fα (F )⎯⎯→ fα (x) for each α ∈ J . Now,

−1 −1 fα (Oα ) ∈ S and if x ∈ fα (Oα ) , fα (x) ∈ Oα ∈ fα (F) and thus fα (Fα ) ⊆ Oα for some

n F ∈F . Hence f −1 (O ) ⊇ F and thus f −1 (O ) ∈F . Since f −1 (O ) is a base α α α α α α ∩ αi αi i=1

member for τ I , it follows that base members containing x belong to F .

HenceF ⎯⎯→τ I x .

In particular, when X = ΠX α is a product set and fα = Pα are projection maps , τ I is

called the product topology for X. According to Proposition 5.1 τ I is the coarsest

topology for X such that each projection map is continuous.

18 CONCLUSION

A discussion of filters and their applications to the theory of general topology has been presented. These ideas have been primarily developed by European mathematicians, beginning with the work of H. Cartan and recorded in the various texts written under the pseudoname “ N. Bourbaki.” The Moore- Smith “ net convergence” has been more widely taught in American universities. Both are used to characterize topological concepts in more abstract spaces.

19 LIST OF REFERENCES

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Bourbaki, N.(1940), Topologie generale. Chapitre 1 et 2. Actualites Sci. Ind., 858

Cartan, H.(1937a), Theorie des filters, compt. Rend., 205, 595-598.

Cartan, H.(1937b), Filtres et ultrafiltres, Compt. Rend., 205, 777-779.

Kowalsky, H. J.(1961), Topologische Raume, Basel-Stuttgart (Topological Spaces, Academic Press, New York, 1964 ).

Preuess, Gerhard, Foundations of Topology, kluwer acod. Pub. , Dordrecht, 2002.

Root, R.(1914), Iterated limits in general analysis, Am. J. Math., 36, 79-134.