• f(x) » y to mean y Î f(x),
• f(x) º { y Î Y | (x,y) Î f }

Because relations are often as awkward in mathematics as in life, it is convenient to consider two ways in which a relation may be encoded as a function. Suppose then, that f : X«Y is some relation. The 'image' of any given x can be any subset of Y. So the first function we can associate with f is already implicit in the notation used above

Countability and [CH]

If two sets X and Y can be placed into 1-1 correspondence with each other, we say they have the same cardinality, written |X| = |Y|. If, in addition, Y Í Z for some set Z, we write |X| £ |Z|. Regarded as an entity in its own right, we call |X| a cardinal, which may be either finite (if X is finite) or infinite. In addition, if |X| £ |N|, we say that X is countable; otherwise it is uncountable.

The sets Æ, N, Z, and Q are countable, while I, R, and C all have the same uncountable cardinality. It is standard practice to abuse notation and write n to denote both the finite ordinal n, and the finite cardinal |n|. We frequently write À₀ for |N|, and c for |I| º |R| º |C|.

Given any cardinality À = |X|, the cardinality 2^À is defined to be |ÃX|. It is easy to show that |N| < |R| £ |ÃN|. The continuum hypothesis [CH] is the assertion that no cardinality exists strictly between À₀ and . This is independent of the axioms of standard set theory [ZFC], and its use in a topological proof (by way of its implication that c º) is signalled by the insertion of the symbol [CH] either at the beginning of the proof, or at the point where it is used.

Ordinals

The collection of sets known as ordinals is defined recursively using three rules.

• The empty set Æ is an ordinal.
• Given any ordinal a, the set a⁺ º a È { a } is an ordinal.
• If S is any set of ordinals, then is an ordinal.

This immediately tells us that the natural order on ordinals, set inclusion, a £ b Û a Í b), is likely to have some special properties. In fact, one of the key factors that make ordinals so important is that they are well-ordered. That is, given any set S of ordinals, S contains a smallest member.

Like all sets, ordinals have cardinality; they are either finite or transfinite. We shall be particularly interested in countable ordinals. The three rules given above are sufficient to define the set of all countable ordinals, provided we make the extra stipulation in rule (3) that the set S must itself be countable. That is: if S is any countable set of countable ordinals, then is again a countable ordinal (this has nothing to do with ordinals per se. Any countable union of countable sets is countable). Because of our convention that n denotes both an ordinal and its associated cardinal, we can regard N as a set of ordinals, whence its union is also a ordinal, which we denote w or w₀. Since n⁺ º n È {n}, we observe that n Ì n⁺ for every n Î N, so that N and w are actually the same set.

We observe, but don't need to go into the detail, that it is possible to define arithmetic on ordinals, much as we define it on integers. For finite ordinals there is no distinction between the two interpretations, but things are very different for transfinite ordinals, and we shall generally avoid ordinal arithmetic in our web-pages. It is worth noting, however, that certain basic identities hold true. In particular, it is always true that a+1 º a⁺, a+a º 2a , and so on. But beware that we cannot reverse the order of addition. Thus, while w+1 º w⁺ and w +2 º w⁺⁺, and so on, we actually have n+w º w for every n < w .

Since w is an ordinal, we can consider w+1, w+2, etc., and eventually we can construct their limit, w+w º 2w. Proceeding in the same way, we can find the limit of 3w, 4w, etc. This is the ordinal ww º w². Using the same idea, we successively create ever larger ordinals:

and since this list is itself countable, we can take the union of all it's members to produce w -to-the-w-to-the-w -to-the-... (w times!).

We need to distinguish three types of ordinal. Our recursive definition bottoms out with Æ, and this ordinal is the only member of its type. Secondly, there are ordinals created as successors of other ordinals. They are called successor ordinals. But some non-zero ordinals cannot be created as successors - for example, there is no ordinal immediately preceding w. If l is an ordinal for which it is possible to find a set of (necessarily smaller) ordinals S, where l º , but l Ï S, then l is called a limit ordinal. The reason for this terminology will become clear when we consider order topologies.

Each of the ordinals described above is countable, but this does not mean that all ordinals are countable. There are uncountably many distinct countable ordinals - consequently their union, denoted w₁, must necessarily be uncountable as well. This ordinal, w₁, is the first uncountable ordinal, and plays an important role in some of the topology which follows.

One final warning before we leave ordinals: It is easy to show that there is no such thing as the "set of all ordinals". If there were such a set, S, we could take its union to generate a new ordinal u. Now u Ì u⁺, and by definition u⁺ Î S. Consequently, we could derive the impossible statement u Ì u⁺ Í u. This tells us automatically that there is no upper bound on the cardinality of ordinals. Given a set X, we can always find an ordinal a for which |X| £ a. It is tempting to suppose that we can always find an a with exactly the same cardinality as X, but in fact this cannot be proven without the use of [AC]. It is a symptom of the subtlety of ordinals that always being able to find such an a (the well-ordering principle) is actually equivalent to [AC], and cannot be established nor refuted in standard [ZF] set theory without [Choice]. In our work, it is convenient to take this as the definition of [AC] (the axiom of choice).

Order isomorphisms and embeddings

When we say that n+wº w , we are actually making a statement about two ordered sets having the same structure. In general, if f : á X, £_Xñ ® á Y, £ _Yñ  is a function from one ordered set to another, it can be said to respect the two orders simultaneously provided

This relationship forces f to be injective. For if f(a) º f(b) we have both f(a) £_Y f(b) and f(b) £ _Y f(a), whence we also require both a £_X b and b £_X a, and hence a º b. Any function with this property is called an order embedding of á X, £_X ñ into áY, £_Y ñ. It essentially says that X can be regarded as a subset of Y, with its order inherited from that of Y. If f is not just an injection, but actually a bijection, we call it an order-isomorphism, and consider X and Y to be essentially the same ordered set.

Well-ordered subsets of the real line are countable

Recall that a set is well-ordered if each of its subsets contains a smallest element. The open interval (0,1) shows clearly that R is not well-ordered. Nonetheless, it contains many subsets that are well-ordered with respect to the natural order in R. An obvious example is the set N itself. More significantly, f_m(n) º m + n/(n+1) order-embeds N within the half-open interval [m,m+1), and putting all of these copies of w together gives us w copies of w - in other words w² - as a subset of [0,¥). Since x = 2.tan^¬(x)/p is an order-isomorphism of [0,¥) onto [0,1), we can contract this copy of w² to occupy only the interval [0,1). We can now piece together w copies of w² to obtain an order-embedded copy of w³ - and clearly we can repeat the process as often as we like to obtain a copy of wⁿ, for any n Î N.

This demonstrates that R can contain some fairly complex well-ordered subsets. But all of the examples we've discussed are countable - while R is uncountable. Is it possible to order-embed w₁ (or any other uncountable ordinal) within R? We already know enough to give the answer: No.

To see why, assume that we've managed to find some order-embedding f : b ® R of some ordinal b within R. We're going to confine our attention to the subset S of b defined by S º { a Î b | a⁺⁺ Î b } (which could be empty, but only if b < 3). Clearly, if b is infinite, then b and S have the same cardinality; since b \S has at most two elements (the two elements 'furthest to the right' in b ). So it's enough to prove that S is countable. To do this, we associate each a in S with an open interval U_a Ì R, where

U_a º

Restricting attention to S ensures simply that the end-points of U_a are defined. Now recall that f is an order-embedding, whence ... < f(a ) < f(a⁺) < f(a⁺⁺) < .... So U_a contains precisely one element of imf, namely f(a⁺). In particular, of course, this means that U_aisn't empty. What's more, the end-points of U_alie exactly half-way between f(a⁺) and its neighbours, so none of the U_a's overlap.

So far, then, we've managed to associate every a in S with a non-empty open interval in R, and these intervals don't overlap. Now we use the fact that every non-empty open interval in R contains infinitely many rationals. We don't need all of them; just one will do. For each interval U_a, we pick a single rational number q_a Î Q Ç U_a. Doing this establishes an injection S ® Q, whence |S| £ |Q| º À₀.

This proves that S must be countable, and so, therefore, is b.

Sequences, directed sets and nets

One of the principle applications of partial ordered sets is in the definition of sequences and convergence. For example, we might define the continuity of a function ¦: R® R by saying that ¦(x_n) ® ¦(x) whenever x_n ® x. This definition introduces two sequences: áx_n ñ and á ¦(x_n) ñ . A sequence in a set X is essentially just a function x:w ® X, except that we normally write x_n instead of x(n) and á x_nñ instead of x. But why should we restrict ourselves to the use of w as the indexing set for the x_n's, when we can clearly generalise the idea to cater for other posets just as easily? Clearly, there is little point in generalising definitions simply for the sake of doing so. But we shall see below that this is one area where a generalisation is actually necessary - it simply isn't possible to define convergence in settings more general than R if we restrict ourselves to sequences.

The properties of w that are so useful in defining convergence are that no matter how far along a sequence we've gone, we can always go further, and no matter how much further we go, we're always heading in the same general direction. More generally, a poset á L £ ñ is directed provided every pair {x, y} Í L is bounded above in L. That is, given any x and y in L, we can find a zÎ L for which both x £ z and y £ z. Like w, directed sets encapsulate the idea that we can always keep moving further along the list of values, but does so in a more general manner, since we no longer require L to be countable, nor do we require £ to be a total order.

Having replaced w with a directed set, we now replace sequences. A net in the set X is a function net:L ® X, where L is some directed set. Borrowing the notation of sequences, we normally write x_l rather than net(l), and write áx_lñ to denote the net as a whole.

Topological terminology and notation

A topology on X is a collection Á of subsets of X (called the open sets of the topology) which includes both X itself and the empty set, Æ , and such that all finite intersections, and all (finite or infinite) unions, of members of Á are also members of Á. If U is an open set, the complement X\U is a closed set. If x Î U Í A, then A is a neighbourhood (nhood) of x (some authors require A to be open; we prefer to call such a set as an open nhood).

If B Í Á has the property that every open set is a union of sets in B, we call B a basis for Á . If S Í B has the property that every set in B can be written as a finite intersection of sets in S, we call S a sub-basis for Á . In order to specify Á, it is clearly sufficient to specify any sub-basis, since Á itself can then be defined as the smallest collection of open sets containing Æ, X, and S, together with all sets obtainable by first taking finite intersections and then arbitrary unions.

The pair áX,Áñ is a topological space, whose underlying set is X. It is usual (except where ambiguity might otherwise arise) to abuse notation and write X to represent both the set X and the topological space á X,Á ñ , whenever the space in question has a standard topology (i.e., one which has historically come to be associated with it in the literature). Unless the context requires otherwise, the term space shall be taken to mean topological space.

It is entirely possible for a subset of X to be simultaneously open and closed; we call it clopen. The sets X and Æ are always clopen in X. If any other subset U Ì X is clopen, so is its complement, and we say that the pair á U, X\U ñ disconnects, or is a disconnection of, X, and that X is disconnected. If X cannot be disconnected, it is connected. The spaces R and I are connected; the spaces N and Q are disconnected.

Closure, dense sets, separability

Suppose that A Í X, and that every open set that contains x also contains at least one point from A. We say that x is adherent to A, or that it is a closure point of A. Obviously, if x is itself in A, this condition will automatically be satisfied. If, however, we can always find a point of A other than x, we say that x is an accumulation point of A. The set of all points adherent to A not only contains A, but is actually the intersection of all the closed sets that contain A. It is a closed set itself, and is called the closure of A in X, written (depending on author and context) Cl_XA or , or simply .  Reflecting the definition of open sets, it is easy to show that any intersection of closed sets, and any finite union, is again closed.

The boundary of A, written ¶_XA (or simply ¶A) comprises those points that lie in the closure of both A and its complement. In other words, x Î ¶A precisely when every open set containing x contains both points in A and points not in A. For example ¶_RQ º R and ¶_RN º N. By construction, boundaries are the intersection of two closed sets, and are therefore closed in their own right.

If  = X, we say that A is dense in X. This means that no matter what open set you choose containing a point x Î X, it will also contain points of A. Essentially this means that much of the structure of X is actually determined by the structure of A. In particular, if we can find a dense subset of X that is countable, we say that X is separable. For example, the space R is separable, because Q is both dense in R and countable.

Subspace topology

A subset A Í X inherits a natural topology from that on X, namely

Unless stated otherwise, subsets are always assumed to carry this subspace topology, and are then called subspaces. Again, we typically abuse notation and write subspace relationships as if they were standard set-theoretical relationships, e.g. Q Ì R means both "Q is a subset of R" and also "Q is a subspace of R".

Continuous functions

If á X,Á _Xñ and á Y,Á _Yñ are both topological spaces, a function f Î Y^X is continuous (cts) provided inverse images of open sets are open, i.e., U Î Á_Y Þ f^¬(U) Î Á_X. In spaces like R which carry a metric (some measure of the distance between points), it is traditional to define continuity in terms of so-called "epsilon-delta" criteria. The two definitions are equivalent for functions defined on such spaces. The topological definition has the advantage, both of being simpler, and also of applying to spaces for which no metric can meaningfully be defined.

We are frequently interested in the set of all continuous functions between one space and another. The set of continuous functions X®Y is denoted C(X,Y). In the specific case where Y º R, and we are considering real-valued functions, we drop the reference to Y and write C(X) º C(X,R). In addition, when discussing C(X) it is often useful to consider the subset of continuous functions which are bounded on X. These form the set C*(X).

Separation axioms

A point or subset A of X can be (topologically) separated from another point or subset B of X provided there is some open set U which contains A and is disjoint from B. If A and B can be mutually separated (i.e. each can be separated from the other), we say simply that they can be separated. In addition, A and B can be functionally separated if there is some continuous function ¦ :X® [0,1] such that ¦ takes the value 0 everywhere in A and the value 1 everywhere in B. In this situation, we observe that the half-open intervals [0,1/2) and (1/2 ,1] are open in I, whence their inverse images are open in X. Since they contain A and B respectively, functionally separated points and sets can also be topologically separated.

In general, topological arguments require there to be 'sufficient' open sets, or else 'sufficient' functions, for us to distinguish between points of the underlying set. The availability of 'enough' open sets and functions is expressed in terms of separation axioms. These are typically known both by a name, and also an abbreviation of the form T_n where n is an integer (for separation) or half-integer (for functional separation) - this is a historical accident reflecting the use of the German word Trennung (separation) in many of the subject's seminal early papers, and the fact that functional separation was introduced after the basic T_n classification had come into common parlance. For our purposes, the most important of these axioms are listed here, where x and y are arbitrary points of X, and A, B Í X are closed subsets.

Tychonov spaces are also known as completely regular spaces.

Compactness

The separation axioms summarise the requirement for 'enough' open sets. At the other extreme we also sometimes need to know that not all of these sets 'need to be taken into account' in a proof.

If S Í 2^X and  º X, we call S a cover of X. If R Í S is also a cover, we call it a subcover of the cover S. If every set in S is open (so that S Í Á ) we call S an open cover. Finally if S is finite or countable we call it a finite (or countable) cover.

The space X is compact provided every open cover has a finite subcover. Compact spaces are very important in general topology for a variety of reasons. Closed subsets of compact sets are themselves compact, and conversely any compact subspace of a Hausdorff space is closed. One of the motivating forces behind the definition of compactness was the discovery of a famous example, viz. any closed and bounded subset of R is compact.

Suppose that K is compact and f :K® Y is continuous. If S is any open cover of imf, consider the sets f^¬(U) where U ranges over the members of S. Because U is open and f is continuous, each of these inverse images is open, and taken together they form a cover of K. By compactness they have a finite subcover, and if we look at the U's associated with this finite subcover we see that they must themselves form a cover of imf. In other words, the continuous image of a compact set is itself compact.

Quotient and product topologies

Suppose that áX,Á ñ is a space, Y is simply a set (with no topology as yet) and f :X ® Y is some surjection of X onto Y. We want to find some topology for Y which makes f continuous. An easy solution would be to impose the so-called indiscrete topology, whose only open sets are Æ and Y itself. This would certainly make f continuous, since the only inverse images we'd need to consider would be Æ and X, which are both open in X. But this isn't particularly adventurous, and besides we haven't used f itself in any way. Instead, we want Y to have as many open sets as possible, without violating continuity of f. The obvious choice is the collection of sets

This is indeed a topology, called the quotient topology induced on Y by f . When equipped with this topology, we call Y a quotient space of X, and f a quotient function.

The quotient topology reflects a situation in which we want to define a topology on an image space in order to make a function continuous. Creating a product topology requires the opposite. We will be given a set of functions from a common domain onto a selection of image spaces, and have to create the appropriate topology for the domain.

Suppose then X_i is a collection of spaces, indexed by some parameter set I. The product set P ºcan be regarded as the set of all functions x: I ® such that x(i ) Î X_i for every i Î I . Given this interpretation, the function p_i(x) = x(i) is called the i^thprojection map, and the space X_i the i^thfactor space.

What topology can we define on P that ensures that each of the p_i's is continuous? Once again, the easy answer is in many ways the least interesting. If we assign P the discrete topology, in which every subset is open, these will include all the inverse images of open sets in the various factor spaces.

Instead, we'll try to find the topology with fewest open sets. One possibility is to take as sub-basis all sets of the form , with U_i open in X_i. This yields the so-called box topology on P, but is generally not the topology we seek, because there is an even smaller topology whenever I is infinite. This topology, the (Tychonov) product topology, is obtained by taking as sub-basis the same product sets as for the box topology, except that we require U_i º X_i for all but finitely many of the U_i.

In addition to making every p_i continuous at the same time, but with the smallest collection of open sets, the product topology has two interesting and extremely useful properties, the second of which is regarded by many as the single most important property in general analytic topology:

• A non-empty product space is T₂ if and only if every factor space is T₂
• A non-empty product space is compact if and only if every factor space is compact

Convergence

In the real line, we know that any point can defined as the intersection of countably many open sets containing it - we can use the open intervals U_n º (x-1/n , x+1/n ) where n ³ 1. By choosing points x_n Î U_n\{x} we obtain a sequence in R\{x} converging to x. In more general spaces this process cannot be guaranteed to work. Frequently, the concept of arithmetic plays no role, and the intervals U_n cannot be defined.

The key goal that convergence must achieve if it is to be meaningful is related to the concept of closure. By definition, x is an accumulation point of A Ì X is every nhood of x contains points of A\{x}. In other words, we can find points of A "arbitrarily close to" x. Intuitively, there should be some 'sequence' in A\{x} that converges to x. However, it is sometimes possible to choose x Ï A which is an accumulation point of A, even though x is not in the closure of any countable subset of A. Consequently, whenever we have a countable subset of A, we can find an open set which separates x from that subset. The necessity that sequences be countable therefore means that they are not enough to define convergence in arbitrary situations. Instead, we turn to nets. Recall that a net is a function L ® X for some directed set L .

Suppose áx_añ is a net L ® X. We say that áx_añ converges to x Î X, and write x_a ® x, if, for every nhood U of x, there is some l Î L such that (a ³ l) Þ x_a Î U. We sometimes say that á x_a ñ is eventually in U.

In addition, it is sometimes useful to consider nets with convergent subnets. Since different subnets may converge to different points, we need to distinguish this concept from convergence itself. We call x a cluster point of áx_añ if, for every nhood U of x, and every l Î L, there is some a ³ l for which x_a Î U.

Cluster points provide a surprisingly powerful concept. In fact, the following are equivalent statements for any topological space X:

• X is compact
• every net in X has a cluster point

Function space topologies

If X and Y are spaces, we can endow the space Y^X with various topologies, all of which are 'natural' in one context or another.

If we disregard the topology on X, and think of it simply as an index set, we can define |X| copies of Y (e.g. Y_x º Y for each x Î X), and create their set-theoretical product. This is just the space Y^X, whence we already have a natural candidate for the function space topology - the product topology on F = Y^X. In the context of function spaces, this is called the pointwise topology because it has the following useful property:

Suppose á¦_añ is a net, and ¦ a function, in F, and that F carries the pointwise topology.
Then ¦_a ® ¦ in F if and only if ¦_a(x) ® ¦(x) in X for every x Î X.

Worked Example: Pointwise convergence in R®R

For each finite subset F Ì R let c_F be the characteristic function of F (that takes the value 1 on F and 0 elsewhere). The functions c_F are all members of R® R, and are indexed by the finite sets F. The finite sets themselves are directed under set inclusion, because two finite sets are bounded above by their union, which is also finite. Consequently, the c_F's form a net ác_Fñ in R^R. Does this net converge in R^R when we equip this space with the pointwise topology? Because we're using the pointwise topology, we simply have to decide if the net ác_F(x)ñ converges in R for every x Î R. Remember - the values c_F(x) are just numbers; there's nothing complicated about them.

Recall the criterion for convergence: we say that x_a ® x, if, for every nhood U of x, there is some l Î L such that (a ³ l ) Þ x_a Î U.

We'll demonstrate that c_F(x)® 1 for every x. This is extremely simple. If we put l = {x}, the definition of a characteristic function tells us that c_l(x) º 1. We also know that the F's are ordered by set inclusion, so whenever a ³ l , we actually have l Í a , whence x Î a. Consequently c_a(x) º 1, and the c_a(x)'s are actually eventually constant with value 1.

This tells us something rather unexpected. Each of the functions c_F takes the value 1 at just finitely many points, and the value 0 at all uncountably many other points of R. Yet their limit in the pointwise topology is the function 1: R® R which is identically 1 across the whole of R.

This seems unintuitive, to say the least, and actually reflects the fact that the pointwise topology has relatively few open sets. There aren't enough 'small' open sets to separate the function 1 from the set of c_F's. In a way this isn't too surprising - the pointwise topology makes no reference whatsoever to the topology on X, but regards it simply as a standard index set. It makes no difference if X is compact, Hausdorff, or even indiscrete, with only two open sets. If we want to get some more open sets, we ought to pay more attention to the structure of X itself.

The compact-open topology

If x Î X and U is open in Y, define a set of functions (x, U) º { ¦ Î Y^X | ¦ (x) Î U }. It can be shown that these sets form a sub-base for the pointwise topology on Y^X, whence it could be called the (point, open)-topology. This suggests a generalisation of the topology, where we replace x with a compact subspace K. (It is a frequent observation that compact sets behave in many ways as if they were 'extended points', so this is an obvious topological technique to apply.) The resulting compact-open topology has as its sub-base the sets

Because there are typically more compact sets than points in X, the compact-open topology typically has more open sets than the pointwise topology. Nonetheless, many of the basic properties of Y^X are determined solely by the equivalent properties on Y. Thus

• if Y is T₂, so is Y^X in the compact-open topology
• if Y is T₃, so is C(X,Y) Í Y^X in the inherited compact-open topology

Order topologies

Of particular relevance for our purposes we observe that any poset carries a natural topology. If á X,£ ñ is a poset, with a Î X, we define the open interval (a, b) to be the set (a,b) º { x Î X | a < x < b }. Depending on the nature of the order, this interval may be empty. The order topology on X is the smallest topology that contains all the open intervals. In practice, this means forming all the sets that are finite intersections of open intervals, and then taking all of the unions of these intersections. Remembering to add both Æ and X itself, the collection formed in this way will be the topology we seek. In particular, the sets ¯_Xa º {xÎ X | x < a } and ­_Xa º {xÎ X | a< x} are both open. The standard topologies on N, Z, Q, and R are all (defined to be) their order topologies. The topology on C is not an order topology - in fact, there is no possible ordering of the complex field which generates the standard topology. Instead, it carries the product topology of R ´ R.

Given any ordinal a, it has an order topology, which we take to be its standard topology. These topologies are consistent with the subset relationships between ordinals. In other words, if a Î b , we know that a Ì b , and indeed the order topology on a is the subspace topology inherited from the order topology on b . In particular, when we consider it in its guise an a topological space carrying the order topology, we (somewhat confusingly) define W₀ to be the first uncountable ordinal (i.e. the set of all countable ordinals) and W º W₀⁺ (the countable ordinals together with the first uncountable ordinal, w₁).

We can say quite a lot about the topology of ordinals. First of all, they are always Hausdorff (easy exercise). We observe moreover that {0} º 1 is open in 1 (X is always open in X), and is also open in every other ordinal, since {0} º ¯_a1. Similarly, if b is a successor, there is some a for which b º a⁺. Since a⁺ is the only ordinal satisfying a < a ⁺ < a⁺⁺, we see that {b} is an open set in its own right. Consequently, all non-limit ordinals are isolated as points in every ordinal that contains them. In particular, every finite ordinal is isolated in w, whence w º N is discrete in the standard topology.

Finally, let's consider an arbitrary limit ordinal, l . Like any ordinal, l is a directed set since any pair {a , b} is bounded above by max(a , b). Consider the space l⁺ - the smallest ordinal containing l . The points of l⁺ \{l} can be considered a net á ord_a ñ , indexed by themselves, i.e. ord_a º a . Any nhood of l in l ⁺ necessarily contains an interval of the form (a ,l] for some a < l, whence it contains every ord_b for b ³ a . Therefore, the net l Ì l ⁺converges to the point l Î l ⁺, and l really is the limit of the ordinals that precede it.

Worked Example: If l is a limit ordinal of any cardinality, then l⁺ is compact

Let l be some limit ordinal, and suppose S is an open cover of l⁺. At least one of these open sets has to contain l itself, which means it contains a basic interval of the form (a ,l⁺) º (a ,l ]. Let a₀ be the least ordinal such that (a₀,l] is contained in some U Î S. We'll call this open set U₀. If a₀ isn't 0, let a₁ be the least ordinal such that (a₁,a₀] is contained in some U₁ Î S. We keep repeating this selection process until we reach a stage when a_n º 0. Such a stage must be reached, since we'd otherwise have an infinite decreasing sequence á a₀, a₁,... ñ , which would constitute a set of ordinals with no least element - contrary to well-ordering. These n+1 members of S cover the whole of l⁺, with the possible exception of 0 itself. We need to add at most one more member of S to cover 0 as well. Consequently, S has a finite subcover, and l⁺ is compact, as claimed.

In particular, W is compact. Moreover, since closed subsets of compact sets are themselves compact, every successor ordinal a⁺ = [0,a] is compact in the order topology. Ironically, it is a fact that every compact subset of a T₂ space is closed. Since l is the limit of its predecessors in l⁺, that set of predecessors cannot itself be closed, and so cannot be compact. But that set is l itself.

Consequently, ordinals have this very strange dual behaviour. Considered as points their properties are essentially the "opposite" of their properties when viewed as spaces. Thus, thought of as a point, a limit ordinal closes off the entire collection of all its predecessors, no matter how many, and leaves l⁺ compact; but viewed as a space, limit ordinals are the only non-compact ordinals.