Kelley topology djvu

For any subset Eofa topological. Terval with the same endpoints. If A denotes the set of rationals in R. Frp '. Examples of topologies 1. Describe the closure and interior operations. Verify that A A is a closure operation. Describe the open sets in the resulting topology. The plane with this topology will be called the radial plane. It is called the simple extension of x over A. There is a subset of R which gives Regularly open and regularly closed sets An open subset G in a topological space is regularly open iff G is the interior of its closure.

A closed subset is regularly closed iff it is the closure of its interior. The complement of a regularly open set is regularly closed and vice versa. There are open sets in R which are not regularly open. If A is any subset of a topological space, then Int Cl A is regularly open.

The intersection, but not necessarily the union, of two regularly open sets is regularly open. Unions, of closed sets!

B; of closed sets in a topological space A' whose union is not clo-ed. The lattice of topologies 1. Tnc intersection of any family of topologies on A' i. Tlie union of two topologies on. But for any family of topologies on X. Thus, the topologies on a fi. The question of whether or not this lattice is complemented has only recently been answered see notc'd- 3H.

The complement of a is an and vice versa. Hence a G, can be written as a decreasing intersection. Much later, sec In a metric space, J I?? For c , you will have to use transfinite induction and 3H. In this and the next section, we present the two most popular ways to describe topologies. If X is a topological space and x e X, a neighborhood hereafter abbreviated nhood of.

The collection of all nhoods of. X is the nhood system at. The next theorem is similar to Theorems 3. A'-c G c A'. Proof is obvious. For A'-b; if V. Then for each ye F, j-g U'. The converse assertion is left to E. Not only have wc lost the linear order in our notion ofclo.

If ,v is close to. Wc can be content with a nhood base. A nhood base at x in the topological space X is a subcolicction taken from the nhood. Obviously, the nhood system at. There are more interesting examples. U is also a nhood of x. For us. In fact, we need consider only the disks of rational radius to obtain a nhood base at x, so each point in a metric space has a countable nhood base. A topological space in which every point has a countable nhood base is said to satisfy the first axiom of countability or to be first countable.

Thus every metric space is first countable. We will meet the second axiom of countability in Exercise 5F ; both axioms will be studied in greater detail in Section Notice that this base at x has no set in common with the nhood base described in b , although they both describe the same topology. Sometimes context or general usage make this clear. We turn now to the problem of specifying a topology by giving a collection of basic nhoods at each point of the space.

Note that Ll-d is dropped altogether. The following theorem is used much more often than the corresponding Theorem 4.

Let X be a topological space and for each x e X, let be a nhood base at x. The properties I'-a, J'-b and V-c arc easily verified for basic nlioods. U-b, and U-c for nhoods. We will proceed to the converse. Suppose a collection satisfying K-a, V-h and V-c has been prescribed at eacli. LC 6 then for. Thus C, r. Moreover, it is clear that, at each. There is a useful alternative to the usual topology on the real line wliich is best described in terms of basic nhoods.

The Sorejenfrey line. Some of its basic properties will be studied in E. It is named after the man who first produced it. We close this section by introducing a concept which depends for its definition on the use of nhoods. The set A' of all cluster points of A is called the derived set of A. Proof From 4. On the other hand, if every nhood of x meets A i. The Sorgenfrey line The following material concerns the Sorgenfrey line, E, introduced in 4. DfvcriKc the c! The Mire plane Let r denote the closed upper half plane [f.

At the point. Verify thtit this gives a topology on F. Compare the topology thus obtained with the usual topology on ihc closed upper half pl. Dc'eribc the closure and interior operations in the. Hereafter, the symbol F will be reserved for the closed upper half plane with the topology described here. This space is often called the Moore plane. We will find consistent use for it as a counterc. Verify that this gives a topology on the plane. Compare this topology with Ihe usual topology on the plane.

X' of the real line other than the origin, Ihc basic nhoods of. X , for til! Verify that this gives a topology on the line. Describ-c the closure operation in the resulting space. This sp. TopoliHjies from nhoods I. Show that if each point. V-d of 4,Z then the collection r - [G cr A' j for each ,x in G. Show that, if is a nhood base at x for each x in a topological space X, then K-a, V-h, V-c and V-d of 4. This is not unusual. In fact, it is a situation to be desired ; spaces without this property are difficult to deal with.

See the discussion in Sections Verify that the sets V f, e form a nhood base at f making R' a topological space.

Compare the topologies defined in 1 and 3. If the definition in 3 is made to apply to continuous functions only, show that the resulting topology on C I is the one induced by the metric defined in 2B.

We will return to the topology in 1, in a more general context, in Section 8 on product spaces. Both the topologies on R' introduced here are treated in the chapter on function spaces. In a metric space X without isolated points, the closure of a discrete set in X is nowhere dense in X.

In any space X, the frontier of an open set is closed and nowhere dense. Conversely, every closed nowhere dense set is the frontier of an open set. In a metric space X, the frontier of an open set is the set of accumulation points of a discrete set. In much the same way, the topology on all of X can be specified, without describing each and every open set, by giving a base for the topology.

A That is, T can be recovered from. A by taking all possible unions of subcollcctions from. A is a base for ,V iff whenever G is an open set in A' and p 6 G. A of all open intervals is a base for the usual topology. More generally, in any metric space M. VJ is a base for the discrete topology on A'. The following theorem is similar to 3. That is, it lists a few properties that bases enjoy and provides the converse assertion ; any structure on a.

Note that no mention is made in this theorem of the topology. If you have a given topology x and want to know whether a particular collection A of sets is a base for x. A is a base for. Suppose, on the other hand. A' is a set and. Let T be all unions of subcolicctions from A. Then any union of members of t certainly belongs to T, so T sati.

But by property b , the intersection of two elements of A is a union of elements of. Thus x satisfies G-2 of 3. I- inally. This completes the proof that x is a topology on X.

Indeed, as the next theorem makes clear, the only real difference between the two notions is that nhood bases need not consist of open sets. The elements of clearly nhoods of x. Moreover, if U is any nhood of x, then. The reduction from topology to base was ac- complished essentially by dropping property G-1 of topologies. The further reduction to subbase is accomplished by dropping G-2 see 3.

Any collection of subsets of a set X is a subbase for some topology on X. Exercise 5 D. Examples of subbases 1. The family of sets of the form — oo, a together with those of the form b, oo is a subbase for the usual topology on the real line. Describe the topology on the plane for which the family of all straight lines is a subbase. Describe the topology on the line for which the sets a, oo , a e R, are a subbase.

Describe the closure and interior operations in this topology. Examples of bases 1. The collection of all open rectangles is a base for a topology on the plane. Describe the topology in more familiar terms. I'or each po'fitivc intcccT n. Describe the closure opcnilion in this space. Describe the interior operation in the resulting space. The scattered Hue We introduce a new topology on the line as follows: a set is open iff it is of the form U u 1' where i' is an open subset of the real line with its usual topology apd K is any subset of the irrationals.

Call the resulting space S. S is a topological space. Describe an efitcient nhood base at a the rational points b the irrational points in S. Bases for the closed sets A base far the closed.

Second countable and separable spaces A space A' is second coiirnahle iff A' has a countable base. Such a set D is. A separable metric space is second countable. Note that this requires the axiom of choice. The Sorgenfrey line K 4,6 is first countable and separable; we will sec later that it cannot be second countable. Materhd on separable and second countable spaces will be developed in the text in Section Chapter 3 New Spaces from Old 6 Subspaces A subset of a topological space inherits a topology of its own, in an obvious way.

This topology and some of its easily developed properties will be presented here. The fact that a subset of X is being given this topology is signified by referring to it as a subspace of X. Any time a topology is used on a subset of a topological space without ex- plicitly being described, it is assumed to be the relative topology. This natural and convenient convention has the result that any adjective which can be applied to topological spaces e.

We are not saying that if a space has a particular property, then every subspace of that space has the same properly; see 6B. The integers, as a subspace of R, inherit the discrete topology.

Each of these examples is a special case of the general rule : if X is metrizable and A c: X, then the relative topology on A is generated by the restriction of any metric which generates the topology on X. The proof of this will be made easy by the next theorem, so it is left to Exercise 6C. By part a , the usual topology on A is generated by the usual metric on A. The proof is easy. The open sets in a subspace A of X are the intersections with A of the open sets in X.

Most, but not alL of the related topological notions are introduced 4t 42 New spaces from old 16 into. If A is it subspace of a topological space X.

The following e. Then Int. Examples of subspaces 1. Recall that A denotes the slotted plane 4C. We will let B denote the radial plane 3A. The relative topolog ' on any circle in the plane as a subspace of B is the discrete topolog '.

Discuss the subspaces of the scattered line S 5C. The rationals, as a subspace of R, do not have the discrete topology. The x-axis in the Moore plane inherits the discrete topolog '. An open set in an open subspace of X is open in X. This need not be true if the sub- space is not open. A similar result holds for closed sets in closed subspaces.

If T is the simple extension over A 3A. Subspaces of separable spaces 1. The Moore plane T 4B is separable see 5F. The x-axis in the Moore plane has for its relative topolog ' the discrete topolog '. Thus, a subspace of a separable space need not be separable. An open subset of a separable space is separable. Subspaces of metrizable spaces If Af is metrizable and N M, then the subspace N is metrizable with the topolog ' generated by the restriction of any metric which generates the topology on M.

The resulting topolog ' on A is the order topology on X and whenever we use the phrase ordered space we mean a linearly ordered set with its order topolog '. An interval in a linearly ordered space is any subset which contains all points between x and y whenever it contains x and y. The usual topolog ' on the real line is the order topolog ' given by the usual order. In I X L with the lexicographic order:.

HA New spaces from old V 4. A suh'Ct of an ordered space has a topology induced by the restricted order ar,J j topolocv inherited from the order topology on the larger space. Show by an example ih;; these two topologies on a subset need not he the same. The basis for our definition i; Theorem 2.

In fact, the reader who restudies Theorem 2. The next theorem provides an alternative, and somewhat surprising, set of characterizations of functions f:X—y Y xvhich arc continuous on all of ,V. V individually. The fourth characterization, although not often used as a test for continuity, is interesting. Thus gof is continuous. This is stated more precisely by the following theorem, and its generalizations in Exercise 7D.

Suppose A and B are open. The proof is similar if A and B are closed. The next theorem says, essentially, that it is not necessary to modify this procedure when dealing with continuous functions.

The proof is left as Exercise 7E. Tlien f is continuous as a map from X to Y iff it is continuous as a map from X to Z. The first is sel-thcorctical : Y will have fewer or. The maps which preserve A' set-thcorctically tind topologically are called homcomorphisms.

In this case, we s;iy A' and Y arc homeomorphic. Various algebraic isomorphisms may be defined in the same formal way. The attempt to unify and. The reader can easily verify the following theorem ; it is a direct con. If X and Y are topological space. Homeomorphic topological spaces are, for the purposes of a topologist, the same.

That i. The reader might profit from thinking, at this point, about the question : is there a set of all topological spaces? To prove two spaces are homeomorphic, one constructs a homeomorphism. To establish that two spaces are not homeomorphic, one must find a topological property possessed by one and not the other. A topological property is a property of topological spaces which, if possessed by X, is possessed by all spaces homeo- morphic to X. First countability, second countability and separability are examples of topological properties which have already been introduced.

We will introduce many more in sections to come. The relations above can be summarized, using transitivity of the homeomorphism relation, as follows : all open intervals in R, including the unbounded intervals, are homeomorphic. Verification of the details passed over here is left to Exercise 7G. In 7G, we will see that we cannot include the unbounded intervals this time. Problems 7A. Characterization of spaces using functions The characteristic function of a subset A oi a set X is the function from X to R which is 1 at points of A and 0 at other points of X.

The characteristic function of A is continuous iff A is both open and closed in X. The analog for topological spaces would be : whenever X can be embedded in 7 and 7 can be embedded in X, then X and 7 are homeomorphic. Find a counterexample. Thus two continuous maps on X to R which agree on a dense subc one v. Sufficient conditions for continuity There arc useful extensions of Theorem 7. A family of subsets of a topological space c called hcallv finite iff each point of he space has a nhood meeting only finitely many elem.

The union of any subfamily from a locally finite family of closed sets is closed. Functions to and from the plane The facts presented here for the plane will be proved in more generality for product spaces in Section S. For each. On the other hand, if q is a function from the plane to any space for each fixed Xr, s R V. C can define a function q,.

Similarly, if 3 'n e R i? We say g is continuous in x iff h ,. The converse to part 2 fails. Homcomorphisms w ithin the line 1. All bounded closed intervals in R arc homeomorphic. The propeny that ever, real-valued continuous function on X assumes us maximcm IS. Thus I is not homeomorphic to R. Topological properties Each of the following expresses a topological property of X : 1. X has cardinal number K, 2.

X is metrizable. Each of the following expresses a property of X which is not a topological property: 6. X is a subset of R. When such a retraction exists, A is called a retract ofX. The unit disk is a retract of the plane. If is a retract of B and 5 is a retract of C, then. Note that lower and upper semicontinuity bear no relation to continuity from the left or right for functions of a real variable ; we are using the ordering of the range of our functions, not the domain.

Most of the results below are stated for lower semicontinuous functions ; they have obvious analogs for upper semicontinuous functions. Every continuous function from X to R is lower semicontinuous. Thus the supremum of a family of continuous functions, if it exists, is lower semicontinuous. This provides a partial converse to part 2. Let C' I be the family of continuously difTercntiablc real-valued functions on I. A linear operator from X to R is called a linear functional.

Here we indulge in the common bad habit of failing to use a distinguLshing notation for the norms on X and F. Show that, in a natural way. Consult any book on abstract algebra for the definition of an algebra. An excellent Introduction to the study of questions of this sort can be found in the book on rings of functions by Gillman and Jerison.

HiX is a group, with composition as the operation. Part 2 effectively disposes of the question asked in the introduction for general spaces X and y. Affirmative answers are available, however, for suitably restricted classes of spaces.

See the notes. In fact, this procedure gives a valid topology, called the box topolotjy. It satisfies our craving for naturality, but is not much used because it is not tame enough, having an ovcr-abundancc of open sets. The definition, given next, of the usual topology used on the product space rectifies this by sharply reducing the number of basis elements. The reader will easily verify that P-a could have been replaced by P-a ' 17, 6. Again, the sets U, can be re- stricted to come from some fixed base in fact, in this case, subbase in A",.

In particular. Hereafter, [][ X, is always assumed to be endowed with the product TychonofT topology if each A', is a topological space. Recall that X is the set of all real-valued functions of a real variable. Then will be a discrete space if and only if A is finite. If A and Y arc topological. In general, an open map need not he closed and vice versa; sec 8A, 9C.

Left as Exercise 8A. The Tychonojf topology is the weakest topology on fJXj. Consequently, the members of a subbase for the Tychonoff topology all belong to t, and hence the Tychonoff topology is contained in r. Necessity of the composition condition is clear since the composition of continuous maps is continuous.

This suffices to show f is continuous. Directly or indirectly, these results lie at the heart of most useful investigations into the properties of product spaces.

As is often the case, a theorem in this case, 8. By Theorem 8. Moreover, Theorem 8. U Theorem. Mimic the proof of 8. B It is one of the remarkable and fruitful results in topology that, with a simple extra condition on the generating collection of maps, any space with a weak topology can be embedded as a subspace of the product of the range spaces. The licart of tlie proof of this theorem lies in the observation that, for each 7 6 A.

Hence, since e is a horneomorphism. Thus the collection! Thus, by Theorem 8. Finally, we will show e is an open map; i.

Since e is one-one. The best known example, the Stone-Cech compactification pX of a Tychonoff space X see Section 19 is typical of the use of 8.

Exercise 8B. Whenever one-point sets in X are closed, a collection of functions which separates points from closed sets will separate points. A space is a Tyspace see Section 13 iff one-point sets are closed. Proof This is a direct consequence of 8. Projection maps 1. Show that the projection of I x R onto R is a closed map.

Separating points from closed sets 1. If X has the weak topology induced by a collection of maps which separates points, this collection of maps need not separate points from closed sets.

Prodiids arc associative and corwinitativc! V, is a topological space for each at g A. Closure and interior in products Let X and V be topological spaces containing subsets A and B, respectively. In the product space A' x Y: 1. Part 2 can be extended to infinite products, while part 1 can be extended only to finite products. Products and the axiom of choice 1.

Show that the axiom of choice is equivalent to the assertion that the product of a nonempty collection of nonempty sets is nonempty, 2. Assuming the axiom of choice, show that each projection map is onto if each factor. The box topology Let A', txe a topological space for each ae A. Compare with 4F3. Work out formulas for the closure and interior of sets in a box product, similar to those given in 8D. Weak topologies on suhspaces Let A' have the weak topology induced by a collection of maps fp.

The identity function from the set X to the space X, will be denoted L. A', x is homeomorphic to the diagonal A in the product space X,. It is also true that there are nonhomeomorphic spaces X and Y such that X x X and y X y are homeomorphic see notes. Sec also 30F. We will be solely concerned in this section with investigating three distinct but equivalent ways of viewing quotient spaces, leaving discussion of strong topologies to Exercise 9H where we show that quotient spaces play a role for strong topologies similar to that played by product spaces relative to weak topologies.

In fact, the conditions we need were given in Definition 8. Since ty is the largest topology making f continuous, t c Xj. Thus Xf c: X and this establishes equality. Let Y have the ptotient topolopy indueed by a map f of X onto Y. X Z is continuous. Necessity is trivial, since the composition of continuous maps is continuous. R fhere is another approach to quotient spaces which yields a great deal of insight. The best approach is to view the necessary construction abstractly, then show it can be used to describe quotient spaces.

Let X be a topological space. You are asked to show that this does give a topology on S in 9B. P is called the natural map or decomposition map of X onto The next theorem says that every decomposition space is a quotient space ; the theorem following that says that every quotient space is homeomorphic to a decomposition space.

See Exercise 9B. It is often of interest, in investigations re- volving around decomposition spaces, to know that F is, in fact, closed. To state the basic result giving conditions on 0 which will make F closed, we introduce the following definition. The natural map P associated with a decomposition space S of X is closed iff 3 is upper semicontinuous. Before moving on to some of the e. It requires nothing but a definition, but represent: probribly the most popular v. Clearly, the last description is the neatest.

If we identify each point 0,. In] Fig. Intuitively, it is clear that the resulting identification space is what one obtains by first rolling the square to obtain a cylinder, as we did in b , then match- ing the ends of the cylinder to obtain a torus Fig. We should mention here that it is clear that any square will produce a cylinder with one pair of sides identified and will give a torus with two pairs of sides identi- fied, as above.

The reason we chose [0, 27t] x [0, 27t] is obvious. It has several interesting properties most of which require combinatorial or algebraic methods to elucidate. In'] x [0. Again the points 0. This can be conveniently represented by arrow. The result, shown in Fig. It is the so-called Klein hotlk. It is a higher-dimensional relative of the Moebius strip. Wc obtain the cone. Wc conclude this section by providing two more methods for generating ne« spaces from old.

Figure 9. We can now employ the construction just accomplished to provide one of the important and interesting ways of generating new spaces. For examples of attachings, see Exercise 9L.

Let F be a closed subset of 1 '. Since g is a quotient map. Let G be an open subset of A' - A. M Problems 9A. Examples of quotient spaces 1. Let 9 be the decomposition of the plane into concentric circles about the origin.

Is the corresponding result for true? Quotients versus decompositions 1. The process given in 9. See 9. Open and closed maps 1. An open continuous map need not be closed, even if it is onto.

A closed continuous map need not be open, even if it is onto. State and prove an analog to 9. Y P he contitniotis, Then theie is ti tpiolieni tnnp q ttf. OiKUit'in ituipx iiiul imuliit'l sixurs Tliefollowittp. V into hoitteotttotpliie sets, sity itll hoineotttoiphieto P, thett X is Itottteontoipltle to?

The fainil. I'or families which cover poittts. T is homcomorpliic to the quotient space obtained by identifying points v and! These results compare with tlie results in 81 on weak topologies and suprema in tf; lattice of topologies.

Covcriiuj spaces Let p be a continuous map of a space A" onto a space X. X is called the base space and jf is the coverit;.

The map p. Every covering projection is a local homcomorphism. The converse fails. A local homcomorphism is an open map. Thus, under a covering projection, the base space is a quotient space of the covering space.

Give conditions under which A' x ' is a covering space of A', with the usual projection map being the covering projection. Attachings 1. Coherent topologies Let.

The topology on X is said to be coherent with. The topology on A' is coherent with. It is sometimes called the weak topology generated by the sets in. Coherent topologies are useful in the study of k-spaces ; see Section A topological space X is first countable or satisfies the first axiom of countability iff each xe X has a countable nhood base. Since the disks about x of rational radius form a nhood base at x in any pseudometric space, the pseudometrizable spaces are all first countable.

They form the most important single class of first-countable spaces. The first axiom of countability has been defined before, in 4. The second axiom was introduced in 5F. Both will be studied in detail in Section Let X and Y be first-countable spaces. This is left as Exercise IOC. A somewhat wider class of spaces can be described using sequences, in fact see the notes , but the following examples show that the basic Theorem Since y does not meet this requirement, no sequence in E can converge to g.

In fl. Pul the order topology 6D on fl. Note that if a is a nonlimit ordinal, ly. Whenever ft is u. Now note that w, efto in this topology. Tlni' sequences fail to describe the topology on ft.

Problems lOA. Sequential com'crgcnec in topological spaces For each of the following spaces, answer these questions : it Which sequences converge to which points?

One of your answers should sliow that finst countability is not necessary in Theorem A' any uncountable set with the cofinitc topology in which the closed. A any uncountable set with the cncowitahle topology, in which the closed sets are A and all countable subsets of X. V any discrete space. A' any trivial space. Topology of first-countable spaces Let X and L be first-countable spaces. Which of the properties above hold for an uncountable set Z with the cofinite topology?

The key to successful generalkation of the notion of sequence, for use in topological spaces, lies in retaining the idea of ordering a collection of points ofX by mapping some ordered set into X, while significantly relaxing the conditions on the ordered sets we will allow.

The following definition has stood the test of time. The first two properties, A-a and A-b, are familiar requirements for an order relation. Note, however, the lack of antisymmetry; a direction need not be a partial order. A-c provides the positive orientation we were seeking for A.

The concept of a net, which generalizes the notion of a sequence, can now be introduced, using an arbitrary directed set to replace the integers. The definition of net convergence is modeled after the definition of sequer.

Let fx;j be a net in a space X. Then x, converges to. V written — x provided for each nhood U of x. This is sometimes said. Note that in both definitions above it is sufficient if we restrict attention to the nhoods in some fixed nhood base at x. For given any nhood V of x. This example should be studied carcfulh : it is the model for most of the proofs, to be given later in this section, of the proper- ties of net.

N of positive integers is a directed set when given its usual ord:: Thus every sequence xj is a net. It is clear that the two definitions ofconvergcr. This illustrates the at fir-'-, strange fact that a subnet can have a much richer index set than the origins; net.

This example is historically important: it is 'xh-. Moore and. Smith to the concept of a net. Then M - Ixf becomes a directed set when ordered bv the relation. A net has y as a cluster point iff it has a subnet which converges to y. Then cp is increasing and cofinal in A, so cp defines a subnet of x;.

It follows that the subnet defined by p converges to y. Suppose a nhood U of y and a point Xq in A are given. If a subnet of x;i has y as a cluster point, so does x;i. A subnet of a subnet of x;i is a subnet of x;i. Then xy is a net contained in E which converges to y. See Example Let f: X Y. Thus for each nhood U of Xq, we can 76 Convergence!

Suppose on the other luind that 7t, x; 7r, x for each a e A. Let be a l asic nhood of x in the product space. Thus if functions on a certain set A to a space A' arc to be studied with pointwisc limits in mind, it is appropriate to consider them as elements in the product space X'' with the TychonofT topology.

There arc other kinds of functional convergence than point- wise. In particular, an ultranct in a topological space must converge to each of its cluster points. For any directed set A. Nontrivial ultranets can be proved to c. Most facts about ultranets arc best developed using liltcrs and ultraliltcrs as a vehicle. We will do this in the next section. Thus, fix, is an ultranct. Examples of net convergence 1. In the ordinal space, recall that co, ello see Example Find a net x,; in SIq which converges to w, in il.

Let M be any metric space. Subnets and cluster points 1. Every subnet of an ultranet is an ultranet. Every net has a subnet which is an ultranet. If x;i is a net in a space X and for each Ac. If an ultranet has x as a cluster point, then it converges to x. The converse fails, even in R x R. Nets describe topologies 1. Nets have the following four properties some have already been mentioned in the text ; a if.

Conversely, suppose in a set X a notion of net convergence has been specified telling what nets converge to what points satisfying a , b , c and d of part 1. We now inlrodiicc j. The result is the theory of filter convergence. A Jilter. F on a set S is a nonempty collection of nonempty subsets ofS with the properties; a if F2 e. A subcollcction. Fq a filter base for. F iff each element oFF contains.

Evidently, a nonempty collection of nonempty sub. F2 arc filters on X, wc say. F, ra. In particular, the set of all nhoods of x e A' is a filter on A', and any nhood base at x is a filter base for This filter will sometimes be called the nhond filler at. Then Y,' is a filter base for a free filter on FI. A filler. F on a topological space X is said to conrerqe to. F is finer than the nhood filter at -v- Wc say has. F clusters at x iff each F e. F meets each U 6 -? I Icnce.

F has. F], Also, it is clear that if. F -» hen. F clusters at. It will be convenient to have the notions of convergence and chistciing available for filter bases; they generalize easily and obviously. A filter base r conrenjes to. Under what con- ditions on A or on the topology will converge to some point? Then 0 although 0 does not belong to every element of ".

The resulting filter contains E and converges to y. Let be the filter of all nhoods of Xg in X. Hewitt : review: j. Kelley, General topology. Login Register. New J. Kelley Springer-Verlag New York John l. Kelley December 6, , was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis.

Read honest and unbiased product reviews from our users. General topology - solution book of john kelley's I have so many difficult in solving problem in General Topology of John Kelley and Topology second edition of James R. Kelley, Mathematics John Leroy Kelley, a member He wrote several papers while a graduate student and completed a thesis in topology under Whyburn in.

John leroy books on amazon. General topology: john l. Kelley, Sam Sloan: Books - Amazon. Amazon Try Prime. Shop by. Skip to Main Content; Sign in. My Account. General topology book 6 available editions General Topology by John L. General Topology. Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis.

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