General Topology

The second method of constructing a new space from a given one depends on dividing the given space X into equivalence classes, each of which is a point of the newly constructed space. Roughly speaking, we “identify” the points of certain subsets of X, so obtaining a new set of points, which is then assigned the “quotient” topology. For example, the equivalence classes of real numbers modulo the integers are assigned a topology so that the resulting space is a “copy” of the unit circle in the plane.

Both of these methods of constructing spaces are motivated by making certain functions continuous. We therefore begin by defining continuity and proving a few simple propositions about it.

There is also a localized form of continuity which is useful.* A function f on a topological space X to a topological space Y is continuous at a point x iff the inverse under f of each neighborhood of f(x) is a neighborhood of x. It is easy to give characterizations of the form of 3.1(e) and 3.1(f) for continuity at a point. Evidently f is continuous iff it is continuous at each point of its domain.

A homeomorphism, or topological transformation, is a continuous one-to-one map of a topological space X onto a topological space Y such that f^–¹ is also continuous. If there exists a homeomorphism of one space onto another, the two spaces are said to be homeomorphic and each is a homeomorph of the other. The identity map of a topological space onto itself is always a homeomorphism, and the inverse of a homeomorphism is again a homeomorphism. It is also evident that the composition of two homeomorphisms is a homeomorphism. Consequently the collection of topological spaces can be divided into equivalence classes such that each topological space is homeomorphic to every member of its equivalence class and to these spaces only. Two topological spaces are topologically equivalent iff they are homeomorphic.

There is no difficulty in extending this definition of product topology to cartesian product of any finite number of coordinate spaces. If each of X₀, X₁, … X_n_–1 is a topological space, then a base for the product topology for X₀ × X₁ × ··· × X_n_–1 is the family of all products U₀ × U₁ × ··· U_n_–1 where each U_i is open in X_i. In particular, if each X_i is the set of real numbers with the usual topology, then the product space is Euclidean n-space E_n. The members of E_n are real-valued functions on the set 0, 1, … n – 1, the value of the function x at the integer i being x_i (= x(i)).

The product topology for the cartesian product of an arbitrary family of topological spaces will now be defined. Suppose we are given a set X_a for each member a of an index set A. The cartesian product

is defined to be the set of all functions x on A such that x_a ε X_a for each a in A. The set X_a is the a-th coordinate set and the projection P_a of the product into the a-th coordinate set is defined by P_a(x) = x_a. Suppose that a topology

is given for each coordinate set. The construction * of the product topology is motivated by the requirement that each projection P_a is to be continuous. In order to attain continuity of the projections it is necessary and sufficient that each set of the form P_a^–1[U] be open, where U is an open subset of X_a. The family of all sets of this form is a subbase for a topology; it is clearly the smallest topology such that projections are continuous. The product topology is this topology. The members of the defining subbase are of the form {x: x_a ε U] where U is open in X_a; they are, intuitively, cylinders over open sets in the coordinate spaces. It is sometimes said that elements of the subbase consist of sets obtained by “restricting the a-th coordinate to lie in an open subset of the a-th coordinate space.” A base for the product topology is the family of all finite intersections of subbase elements. A member U of this base is of the form

a ε F} = {x: x_a ε U_a for each a in F} where F is a finite subset of A and U_a is open in X_a for each a in F. It is to be emphasized that these are finite intersections. It is not true that

is always open relative to the product topology if U_a is open in X_a for each a. The product space is the cartesian product with the product topology.

The projections of a product space into the coordinate spaces have another very useful property. A function f on a topological space X to another space Y is open (interior) iff the image of each open set is open; that is, if U is open in X, then f[U] is open in Y.

2 THEOREM The projection of a product space into each of its coordinate spaces is open.

11 THEOREM If the quotient space X/R is Hausdorff, then R is closed in the product space X × X.

If the projection P of a space X onto the quotient space X/R is open, and R is closed in X × X, then X/R is a Hausdorff space.

PROOF If X/R is a Hausdorff space and

, then P(x) ≠ P(y) and there are disjoint open neighborhoods U of P(x) and V of P(y). The sets P^–1[U] and P^–1[V] are open, and since their images under P are disjoint, no point of P^–1[U] is R related to a point of P^–1[V]. Therefore P^–1[U] × P^–1[V] is a neighborhood of (x,y) which is disjoint from R, and R is closed. The first statement of the theorem is proved. Suppose now that P is open, R is closed in X × X, and P(x) and P(y) are distinct members of X/R. Then x is not R related to y and, since R is closed, there are open neighborhoods U and V of x and y respectively such that no point of U is R related to a point of V. Hence the images of U and V are disjoint, and since P is open they are open neighborhoods of P(x) and P(y) respectively.

12 THEOREM A decomposition

of a topological space X is upper semi-continuous if and only if the projection P of X onto

is closed.

PROOF According to theorem 3.10, P is a closed map iff for each open subset U of X it is true that the union V of all members of

which are subsets of U is an open set. If P is closed,

and D ⊂ U, then V is the required open set and hence

is upper semi-continuous. To prove the converse suppose that

is upper semi-continuous and that U is an open subset of X. Let V be the union of all members of

which are subsets of U. If x ε V, then x ε D ⊂ U for some D in

. Hence by upper semi-continuity there is an open set W, the union of members of

, such that D ⊂ W ⊂ U. Then W is a subset of V and consequently V is a neighborhood of x. The set V is open because it is a neighborhood of each of its points, and it follows from 3.10 that P is a closed map.

If A and B are disjoint closed subsets of X one may define the decomposition

of X whose members are A, B, and all sets {x} for x in X ~ (A ∪ B). The quotient space of this decomposition is sometimes called “the space obtained by identifying all points of A and identifying all points of B.” It is very easy to verify that

is upper semi-continuous, and if X is Hausdorff the relation

is closed in X × X One might suppose that with this simple construction the quotient space might inherit pleasant properties of the space X. Unfortunately, even in this case X may be Hausdorff or satisfy the first or second countability axiom and the corresponding proposition for the quotient space be false.

ACONNECTED SPACES

The image under a continuous map of a connected space is connected.

BTHEOREM ON CONTINUITY

Let A and B be subsets of a topological space X such that X = A ∪ B, and A ~ B and B ~ A are separated. If f is a function on X which is continuous on A and continuous on B, then f is continuous on X. (See 1.19.)

CEXERCISE ON CONTINUOUS FUNCTIONS

If f and g are continuous functions on a topological space X with values in a Hausdorff space Y, then the set of all points x in X such that f(x) = g(x) is closed. Consequently, if f and g agree on a dense subset of X (f(x) = g(x) for x belonging to a dense subset of X), then f = g.

DCONTINUITY AT A POINT; CONTINUOUS EXTENSION

Let f be defined on a subset X₀ of a topological space X with values in a Hausdorff space Y; then f is continuous at x iff x belongs to the closure of X₀ and for some member y of the range the inverse of each neighborhood of y is the intersection of X₀ and a neighborhood of x.

(a) A function f is continuous at x iff and whenever S and T are nets in X₀ converging to x then f ∘ S and f ∘ T converge to the same point of Y.

(b) Let C be the set of points at which f is continuous and let f′ be the function on C whose value at a point x is the member y of the range space which is given by the definition of continuity at a point (more precisely, the graph of f′ is the intersection of C × Y with the closure of the graph of f). The function f′ has the property: If U is open in X, then f′[U] ⊂ f[U]^–. The function f′ is continuous, provided Y has the property: The family of closed neighborhoods of each point of Y is a base for the neighborhood system of the point. (Such topological spaces are called regular. The requirement that Y be regular is essential here, as shown by Bourbaki and Dieudonné [1].)

EEXERCISE ON REAL-VALUED CONTINUOUS FUNCTIONS

Let f and g be real-valued functions on a topological space, let f and g be continuous with respect to the usual topology for the real numbers, and let a be a fixed real number.

(a) The function af, whose value at x is af(x), is continuous. (Show that the function which carries the real number r into ar is continuous, and use the fact that the composition of continuous functions is continuous.)

(b) The function |f|, whose value at x is |f(x) |, is continuous.

(c) If F(x) = (f(x), g(x)), then F is continuous relative to the usual topology for the Euclidean plane. (Verify that F followed by projection into a coordinate space is continuous.)

(d) The functions f + g, f – g, and f · g are continuous, and if g is never zero, then f/g is continuous. (First show that +, –, and · are continuous functions on the Euclidean plane to the space of real numbers. (See also 3.S.))

(e) The functions max [f,g] = [(f + g) +|f – g|]/2 and min [f,g] = [(f + g) – |f – g |]/2 are continuous.

FUPPER SEMI-CONTINUOUS FUNCTIONS

A real-valued function f on a topological space X is upper semi-continuous iff the set {x: f(x) a) is closed for each real number a. The upper topology for the set R of real numbers consists of the void set, R, and all sets of the form {t: t < a} for a in R. If {S_n, n ε D} is a net of real numbers, then lim sup {S_n: n ε D} is defined to be lim {sup {S_m: m ε D and m n}: n ε D} where this limit is taken relative to the usual topology for the real numbers.

(a) A net {S_n, n ε D} of real numbers converges to s relative to iff lim sup {S_n: n ε D} s.

(b) If f is a real-valued function on X, then f is upper semi-continuous iff f is continuous relative to the upper topology , and this is the case iff lim sup {f(x_n): n ε D} f(x) whenever {x_n, n ε D} is a net in X converging to a point x.

(c) If f and g are upper semi-continuous and t is a non-negative real number, then f + g and tf are upper semi-continuous.

(d) If F is a family of upper semi-continuous functions such that i(x) = inf {f(x): f ε F} exists for each x in X, then i is upper semi-continuous. (Observe that .)

(e) If f is a bounded real-valued function on X, then there is a smallest upper semi-continuous function f^– such that f^– f. If is the family of neighborhoods of a point x and Sv = sup {f(y): y ε V}, then .

(f) A real-valued function g is called lower semi-continuous iff –g is upper semi-continuous. If f is a bounded real-valued function, let f_– = –(–f)^– and let the oscillation of f, Q_f, be defined by Q_f(x) = f^–(x) –f_–(x) for x in X. Then Q_f is upper semi-continuous, and f is continuous iff Q_f(x) = 0 for all x in X.

(g) Let f be a non-negative real valued function on X, let R have the usual topology, and let G = {(x,t): 0 t f(x)} have the relativized product topology of X × R. Let be the decomposition of G into “vertical slices”; that is, sets of the form ({x} × R) ∩ G. If the decomposition is upper semi-continuous, then f is upper semi-continuous. (The converse is also true but the simplest proof requires theorem 5.12.)

GEXERCISE ON TOPOLOGICAL EQUIVALENCE

(a) Any two open intervals of real numbers, with the relativized usual topology for the reals, are homeomorphic.

(b) Any two closed intervals are homeomorphic, and any two half-open intervals are homeomorphic.

(c) No open interval is homeomorphic to a closed or half-open interval, and no closed interval is homeomorphic to a half-open interval.

(d) The subspace {(x,y) : x² + y² = 1} of the Euclidean plane is not homeomorphic to a subspace of the space of real numbers.

(Certain of the foregoing spaces have one or more points x such that the complement of {x} is connected.)

HHOMEOMORPHISMS AND ONE-TO-ONE CONTINUOUS MAPS

Given two topological spaces X and Y, a one-to-one continuous map of Y onto X and a one-to-one continuous map of X onto Y: then X and Y are not necessarily homeomorphic. (Let the space X consist of a countable number of disjoint half-open intervals and a countable number of isolated points (points x such that {x} is open). Let Y consist of a countable number of open intervals and a countable number of isolated points. Observe that a countable number of half-open intervals can be mapped in a one-to-one continuous way onto an open interval. I believe this example is due to R. H. Fox.)

ICONTINUITY IN EACH OF TWO VARIABLES

Let X and Y be topological spaces, X × Y the product space, and let f be a function on X × Y to another topological space. Then f(x,y) is continuous in x iff for each y the function f( ,y) whose value at x is f(x,y), is continuous. Similarly, f(x,y) is continuous in y iff for each x ε X, the function f(x, ) such that f(x, )(y) = f(x,y), is continuous. If f is continuous on the product space, then f(x,y) is continuous in x and in y, but the converse is false. (The classical example is the real-valued function f on the Euclidean plane such that f(x,y) = xy/(x² + y²) and f(0,0) = 0.)

JEXERCISE ON EUCLIDEAN n-SPACE

A subset A of Euclidean n-space E_n is convex iff for every pair x and y of points of A and every real number t, such that 0 t 1, the point tx + (1 – t)y is a member of A. (We define (tx + (1 – t)y)i = tx_i + (1 – t)y_i.) Then any two non-void open convex subsets of E_n are homeomorphic. What of closed convex subsets ?

KEXERCISE ON CLOSURE, INTERIOR AND BOUNDARY IN PRODUCTS

Let X and Y be topological spaces and let X × Y be the product space. For each set C let C^b be the boundary of C. Then, if A and B are subsets of X and Y respectively,

(a) (A × B)^– = A^– × B^–

(b) (A × B)⁰ = A⁰ × B⁰, and

(c) (A × B)^b = (A × B)^– ~ (A × B)⁰ = ((A^b ∪ A⁰) × (B^b ∪ B⁰)) ~ (A⁰ × B⁰) = (A^b × B^b) ∪ (A^b × B⁰) ∪ (A⁰ × B^b) = (A^b × B^–) ∪ (A^– X B^b).

LEXERCISE ON PRODUCT SPACES

Suppose that, for each member a of an index set A, X_a is a topological space. Let B and C be disjoint subsets of A such that A = B ∪ C. Then the product space is homeomorphic to the product space . For each fixed topological space X the product X^A is homeomorphic to X^B × X^C and (X^B)^C is homeomorphic to X^B^×^C, all spaces being given the product topology.

MPRODUCT OF SPACES WITH COUNTABLE BASES

The product topology has a countable base iff the topology of each coordinate space has a countable base and all but a countable number of the coordinate spaces are indiscrete.

NEXAMPLE ON PRODUCTS AND SEPARABILITY

Let Q be the closed unit interval and let X be the product space Q^Q. Let A be the subset of X consisting of characteristic functions of points; more precisely, x ε A iff for some q in Q, x(q) = 1 and x is zero on Q ~ {q}.

(a) The space X is separable. (The set of all x in X with finite range (sometimes called step functions) are dense in X. There is also a countable subset of this set which is dense in X.)

(b) The set A, with the relative topology, is discrete and not separable.

OPRODUCT OF CONNECTED SPACES

The product of an arbitrary family of connected topological spaces is connected. (Fix a point x in the product, and let A be the set of all points y such that there is a connected subset to which both x and y belong. Show that A is dense.)

PEXERCISE ON T₁-SPACES

The product of T₁-spaces is a T₁-space. If is a decomposition of a topological space, then the quotient space is T₁ iff the members of are closed.

QEXERCISE ON QUOTIENT SPACES

The projection of a topological space X into the quotient space X/R is a closed map iff, for each subset A of X, R[A]– ⊂ R[A^–]. The projection is open iff R[A⁰] ⊂ R[A]⁰ for each subset A. (^– and ⁰ are the closure and interior respectively.)

REXAMPLE ON QUOTIENT SPACES AND DIAGONAL SEQUENCES

Let X be the Euclidean plane with the usual topology, let A be the set of all points (x,y) with y = 0, and let the decomposition consist of A and all sets {(x,y)} with . Then , with the quotient topology, has the following properties.

(a) The projection of X onto the quotient space is closed.

(b) There is a countable number of neighborhoods of A whose intersection is {A}.

(c) For each non-negative integer m the sequence {(m,1/(n + 1)), n ε ω] converges, in the quotient space, to A. If {N_n, n ε ω] is a subsequence of the sequence of non-negative integers, then the sequence {(n,1/(N_n+ 1)), n ε ω} does not converge to A. (The latter might be called a diagonal of the original family of sequences.)

(d) The quotient space does not satisfy the first axiom of countability.

Note This example is due to R. S. Novosad.

STOPOLOGICAL GROUPS

A triple is a topological group iff (G,·) is a group, is a topological space, and the function whose value at a member (x,y) of G×G is x·y^–1 is continuous relative to the product topology for G × G. When confusion is unlikely all mention of the group operation · and the topology is suppressed, and we say “G is a topological group.” If X and Y are subsets of G, then X · Y is the set of all points z of G such that z = x·y for some x in X and some y in Y. If x is a point of G we abbreviate {x}·Y and Y· {x} to x·Y and Y·x respectively, and Y^–1 is defined to be {x: x^–1 ε Y).

(a) If X, Y, and Z are subsets of G, then (X·Y)·Z = X·(Y·Z) and (X·Y)^–1 = Y^–1·X^–1.

(b) Let (G,·) be a group and a topology for G. Then is a topological group iff for each x and y in G and each neighborhood W of x·y^–1 there are neighborhoods U of x and V of y such that U· V^–1 ⊂ W. Equivalently, is a topological group iff i and m are continuous, where i(x) = x^–1 and m(x,y) = x·y.

(c) If G is a topological group, then i, where i(x) = x^–1, is a homeomorphism of G onto G. For each a in G both L_a and R_a (called the left and right translations by a) are homeomorphisms, where L_a(x) = a·x and R_a(x) = x·a.

It is very important to notice that the topology of a topological group is determined by the neighborhood system of a member of the group. This fact (precisely stated below) permits the “localization” of many notions.

(d) If G is a topological group and is the neighborhood system of the identity, then a subset A of G is open iff for each x in A or equivalently if for each x in A. The closure of the subset A is . (Notice that x ε U·A iff (U^–1·x) ∩ A is not void.)

(e) The family of neighborhoods of the identity e of a topological group has the properties:

(i)	if U and V belong to , then ;
(ii)	if and U ⊂ V, then ;
(iii)	if , then for some , V·V^–¹ ⊂ U; and
(iv)	for each U in and each x in G, .

On the other hand, given a group G and a non-void family of non-void subsets satisfying these four propositions there is a unique topology for G such that is a topological group and is the neighborhood system of the identity element.

(f) Every group, with the discrete topology or with the indiscrete topology, is a topological group If G is the set of real numbers, then (G,+,), where is the usual topology, is a topological group and (G ~ {0},·,) is also a topological group. If G is the set of all integers, p is a prime and is the family of all subsets U of G such that for some positive integer k every integral multiple of p^k belongs to U, then is the neighborhood system of 0 relative to a topology such that (G,+,) is a topological group.

(g) A topological group is a Hausdorff space whenever it is T₀-space. (That is, if x and y are distinct elements there is either a neighborhood of x to which y does not belong or the reverse. Observe that if , then , and if V^–1· V ⊂ U, then V·x ∩ V·y is void.)

(h) If U is open and X is an arbitrary subset of a topological group, then U·X and X·U are open. However, both X and Y may be closed subsets and X·Y may fail to be closed. (Consider the Euclidean plane with the usual addition with X = Y = {(x,y):y = 1/x²}.)

(i) A cartesian product of groups is a group under the operation: (x·y)_a = x_a·y_a for each a in A. The product, with the product topology, is a topological group and the projection into each coordinate space is a continuous open homomorphism.*

Note Bourbaki [1], Pontrjagin [1], and Weil [2] are standard references on topological groups; see also Chevalley [1].

TSUBGROUPS OF A TOPOLOGICAL GROUP

(a) A subgroup of a topological group is, with the relative topology, a topological group.

(b) The closure of a subgroup is a subgroup and the closure of an invariant subgroup is invariant (invariant = normal = distinguished).

(d) The smallest subgroup which contains a fixed open subset of a topological group is both open and closed.

(e) The component of the identity in a topological group is an invariant subgroup.

(f) A discrete (with the relative topology) normal subgroup of a connected topological group is a subset of the center. (For a fixed member h of the subgroup H consider the map of G into H which carries x into x^–1·h·x.)

UFACTOR GROUPS AND HOMOMORPHISMS

Let G be a topological group, H a subgroup, G/H the family of left cosets (sets of the form x·H for some x in G). Then G/H with the quotient topology is a homogeneous space. If H is an invariant subgroup, then G/H is a group, called the factor group or quotient group.

(a) The projection of a topological group G onto the homogeneous quotient space G/H is open and continuous. (Show that the union of all left cosets which intersect an open set U is U·H and apply 3.10.)

(b) If H is an invariant subgroup, then G/H, with the quotient topology, is a topological group and the projection is a continuous, open homomorphism.

(c) The map of the homogeneous space which carries an element A into a·A, where a is a fixed member of G, is a homeomorphism.

(d) If f is a homomorphism of a topological group G into another group H, then f is continuous iff the inverse of a neighborhood of the identity element of H is a neighborhood of the identity of G.

(e) If f is a continuous homomorphism of the topological group G into a topological group J, then the map of G onto f[G], where f[G] has the quotient topology, is a continuous open homomorphism, and the identity map of f[G], with the quotient topology, into J is continuous. Hence each continuous homomorphism may be “factored” into a continuous open homomorphism followed by a continuous one-to-one homomorphism. If f is a continuous open homomorphism of G onto f, then J is topologically isomorphic to G/K where K is the kernel of f.

(f) If J ⊂ H ⊂ G and J and H are invariant subgroups of G, then H/J is a subgroup of G/J, the quotient topology for H/J is the relative quotient topology for G/J, and the map of G/J into G/H which carries A into A·H is continuous and open, and hence (G/J)/(H/J) is topologically isomorphic to G/H.

VBOX SPACES

A base for the box topology for the cartesian product is the family of all sets where U_a is open in X_a for each a in A. Hence the cartesian product of open sets is open relative to the box topology.

(a) Projection into each coordinate space is, relative to the box topology, continuous and open.

(b) Let Y be the cartesian product of the real numbers an infinite number of times; that is, Y = R^A, where R is the set of real numbers and A is an infinite set. With the box topology Y does not satisfy the first countability axiom, and the component of Y to which a point y belongs is the set of all points x such that {a: x_a ≠ y_a} is finite. (Let x and y be points of Y whose coordinates differ for an infinite set a₀, a₁ ···, a_p ··· of members of A. Let Z be the set of all z in Y such that for some k, p| z(a_p) – x(a_p) |/| x(a_p) – y(a_p) | < k for all p. Then Z is open and closed, x ε Z and .)

(c) Prove the results of (b) for the product of an infinite number of connected, Hausdorff topological groups, each of which contains at least two points. Show first that the product of topological groups is, with the box topology, a topological group.

WFUNCTIONALS ON REAL LINEAR SPACES

Let (X,+,·) be a real linear space. A real-valued linear function on X is called a linear functional. The set Z of all linear functionals on X is, with the natural definition of addition and scalar multiplication, a real linear space. It is clear that Z is a subset of the product , where R is the set of real numbers. The relativized product topology for Z is called the weak* or w*-topology (the simple topology). (The space Z is a subgroup of R^X, which is a topological group according to 3.S(i); however, the following results do not require the propositions on topological groups.)

The following propositions characterize w*-dense subspaces of Z and w*-continuous linear functionals.

(a) If f, g₁, …, g_n are members of Z and f(x) = 0 whenever g_i(x) = 0 for each i, then there are real numbers a₁, …, a_n such that f = ∑{a_ig_i: i = 1, …, n}. (Consider the map G of X into Eⁿ defined by (G(x))_i = g_i(x). Show that there is an induced map F (see chapter 0) such that f = F ∘ G.)

(b) Density lemma Let Y be a linear subspace of Z such that for each non-zero member x of X there is g in Y such that g(x) ≠ 0. Then Y is w*-dense in Z. (To show that f ε Y^– it is necessary to prove that for each finite subset x₁, …, x_n of X there is a member of Y which approximates f at each of x₁, …, x_n. Show there is g in Y such that g(x_i) = f(x_i) for each i, i = 1, …, n.)

(c) Evaluation theorem A linear functional F on Z is w*-continuous iff it is an evaluation; that is, iff for some x in X it is true that F(g) = g(x) for all g in Z. (If F is w*-continuous, then for some x₁, …, x_n in X and some positive real numbers r₁, …, r_n it is true that | F(g) | < 1 whenever | g(x_i) | < r_i for each i. Show that, if g(x_i) = 0 for each i, then F(g) = 0.)

Notes The concept of the product topology grew out of the study of sequential convergence relative to the w*-topology. The latter has been studied extensively (see, for example, Banach [1]). There were several awkward situations which arose in this study, which have been somewhat clarified by further topological developments. One might define the sequential closure of a set to be the union of the set and all limit points of sequences in the set, and agree that a set is sequentially closed iff it is identical with its sequential closure. Then it is not hard to see that a set may be sequentially closed relative to the w*-topology but may fail to be w*-closed. This is not a serious criticism if sequential convergence is the object under study. However, the really damaging fact is that the sequential closure of a set may fail to be sequentially closed; that is, sequential closure is not a Kuratowski closure operator. Because of this the machinery of general topology does not apply to the sequential closure operator, and ad hoc arguments are necessary for each conclusion. See Banach [1; 208 ff] for further discussion and examples.

XREAL LINEAR TOPOLOGICAL SPACES

A real linear topological space (r.l.t.s) is a quadruple (X,+,·,) such that (X, +, ·) is a real linear space, (X,+,) is a topological group, and the scalar multiplication, ·, is a continuous function on X × (real numbers) to X. Recall that a subset K of a real linear space is convex iff, whenever 0 t 1 and x and y are members of K, then t·x + (1 – t)·y ε K.

(a) The function which, for a fixed real number a, a ≠ 0, carries each member x of a real linear topological space into a·x is a homeomorphism.

(b) The cartesian product of real linear topological spaces is, with addition and scalar multiplication defined coordinate-wise, and with the product topology, a r.l.t.s.

(c) If Y is a linear subspace of the r.l.t.s. X, then Y, with the relative topology, is a r.l.t.s., and X/Y, with the quotient topology, is a r.l.t.s.

(d) Let K be a convex subset of a r.l.t.s. X and f a linear functional on X. Then f is continuous on K iff, for each real number t, the set f^–1[t] ∩ K is closed in K. (If {x_n, n ε D} is a net in K, converging to a member x of K such that {f(x_n), n ε D} fails to converge to f(x), choose for n in a cofinal subset of D a point y_n on the segment from x_n to x such that f(y_n) is a constant different from f(x).)

(e) If f is a real-valued linear function (that is, a linear functional) on a r.l.t.s. X, then f is continuous iff {x: f(x) = 0} is closed.

Notes The concept of a linear topological space is relatively recent (Kolmogoroff [1] and v. Neumann [1]); it is a notion which grew out of the study of the weak and weak* topologies for a Banach space and its adjoint. Much of the elementary theory of linear topological spaces is a direct application of the theory of topological groups; the results which distinguish the theory from that of topological groups all depend on convexity arguments. (This is a perfectly normal state of affairs; the chief use of the scalar multiplication, which is the only distinguishing feature, is in convexity arguments.) The few results on r.l.t. spaces which occur in the problems of this book do not constitute an adequate introduction to the theory because we do not list the propositions on convexity which are essential to a serious study. The following are suggested as reading references: Bourbaki [3], Nachbin [1], and Nakano [1]. The first of these contains a study of linear topological spaces over a topologized (not necessarily commutative) field.

* If f is defined on a subset A of a topological space, then continuity at points of the closure A^– may also be defined (see 3.D); several useful propositions result.

* A topologist is a man who doesn’t know the difference between a doughnut and a coffee cup.

* Some authors use the term “representation” to mean continuous homomorphism, and the term “homomorphism” to mean a continuous homomorphism which is an open map onto its range.