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Eberlein Compacta
c-18
Eberlein compacta
c-18
145
Eberlein Compacta
Eberlein compacta were introduced to mathematics as compact subspaces of Banach spaces with the weak topology. They have a natural description in terms of spaces of functions with the pointwise topology: a compact space Y is an Eberlein compactum if and only if there exists a compact (Hausdorff) space X such that Y is homeomorphic to a subspace of Cp (X). An equivalent description [2]: a compact space X is an Eberlein compactum if and only if there exists a compact subspace F of Cp (X) such that the functions in F separate the points of X. Yet another characterization [1]: a compact space X is an Eberlein compactum if and only if in Cp (X) there exists a dense σ -compact subspace. The advantage of this criterion is that it describes when a compactum is an Eberlein compactum by a topological property of Cp (X). Arguments involving compact sets of functions play a key role in functional analysis. It turns out that compact parts of Cp (X), where X is compact, have much better convergence properties than Cp (X) itself. In particular, convergence in any Eberlein compactum can be described in terms of usual (countable) convergent sequences, that is, a point x is in the closure of a set A if and only if some sequence (an : n ∈ ω) of points of A converges to x. Such spaces are called Fréchet– Urysohn. On the other hand, for a compact space X, Cp (X) is FrÈchetñUrysohn if and only if the space X is scattered [2]. Since every dyadic compactum of countable tightness is metrizable [E], it follows that every Eberlein compactum which is also a dyadic compactum must be metrizable [2]. Another important property of Eberlein compacta: the closure of any countable subset in an Eberlein compactum is metrizable. Moreover, if A is any subset of an Eberlein compactum X, then the weight of the closure of A in X does not exceed the cardinality of A. Spaces with this property are called monolithic [2]. The importance of monolithicity can be seen from the following simple observation: every separable monolithic compactum is metrizable. Therefore, every separable Eberlein compactum is metrizable. It follows that any Tychonoff or Cantor cube of uncountable weight can serve as an example of a non-Eberlein compactum. In fact, H. Rosenthal proved that every Eberlein compactum with the countable Souslin number is metrizable [2] (see also [3]). This considerably improves the statement on metrizability of separable Eberlein compacta and implies that every non-metrizable compact topological group is a non-Eberlein compactum. Every metrizable compactum is Eberlein, though this is not immediately clear from the deÝnition [2]. The simplest example of a non-metrizable Eberlein compactum is the onepoint compactification of an uncountable discrete space. The class of Eberlein compacta is ìcategoricallyî nice. Indeed, any closed subspace of an Eberlein compactum is an
Eberlein compactum, and the product of any countable family of Eberlein compacta is an Eberlein compactum. We already noted above that the last statement does not expand to uncountable products. Every Hausdorff continuous image of an Eberlein compactum is again an Eberlein compactum. This is a rather deep result, not so easy to prove (see [7]). Monolithicity of Eberlein compacta has another nontrivial corollary. It is known that if a monolithic compact Hausdorff space X is FrÈchetñUrysohn, then X is firstcountable at a dense set of points [2]. It follows that every Eberlein compactum is Ýrst-countable at a dense set of points. Hence, each homogeneous Eberlein compactum X is Ýrst-countable and the cardinality of such X does not exceed 2ω . The Alexandroff double circle is an example of a nonmetrizable Eberlein compactum satisfying the Ýrst axiom of countability at every point. J. van Mill constructed a homogeneous non-metrizable Eberlein compactum [8]. Note that the double circle is the union of two metrizable subspaces (the two individual circles). M.E. Rudin proved that every compactum which is the union of two metrizable subspaces is an Eberlein compactum. However, a compact Hausdorff space which is the union of three metrizable subspaces need not be Eberlein (consider the one-point compactiÝcation of a Ψ -space; the space obtained is not monolithic). Eberlein compacta also enter the picture in connection with the following theorem of Grothendieck: if X is a countably compact space then, for every relatively countably compact subset A of Cp (X), the closure of A in Cp (X) is compact. Note that a subset A of a space Z is said to be relatively countably compact or countably compact in Z if every inÝnite subset of A has a point of accumulation in Z. It turns out that, under these assumptions, the closure of A in Cp (X) is an Eberlein compactum [4, 11, 2]. From the fact that every Eberlein compactum is FrÈchetñUrysohn it follows that every countably compact subset of Cp (X) is compact whenever X is countably compact. However, even more is true: if P is a pseudocompact subspace of Cp (X) where X is countably compact then P is compact and, therefore, closed in Cp (X). A proof of this result is based on the following convergence property of Eberlein compacta: for each point x of any Eberlein compactum X there exists a sequence (Un : n ∈ ω) of non-empty open subsets of X converging to x (D. Preiss and P. Simon [10]). How ìcompactî can Cp (X) be for a compact X? Obviously, Cp (X) is compact only if X is empty. N.V. Velichko showed (see [2]) that Cp (X) is σ -compact if and only if X is Ýnite (then, and only then, Cp (X) is locally compact as well). In this connection the next result is especially interesting: for every Eberlein compactum X, the space Cp (X) is a Kσ δ , that is, Cp (X) can be represented as the intersection of a countable family of σ -compact subspaces of R X
146 [1, 2]. Therefore, for every Eberlein compactum X, Cp (X) is a Lindelöf Σ-space and even a K-analytic space (M. Talagrand [13]). Unfortunately, the Kσ δ property of Cp (X) does not characterize Eberlein compacta; M. Talagrand constructed a compact Hausdorff space X such that Cp (X) is a Kσ δ , and X is not an Eberlein compactum [14]. However, a zero-dimensional compactum X is Eberlein if and only if the space Cp (X, D) of all continuous maps of X in the discrete two-point space D = {0, 1}, in the pointwise topology, is σ -compact [2, 4.6.4]. From Talagrandís theorem (see above) it follows that every linearly ordered compactum which is also an Eberlein compactum must be metrizable. Indeed, it was shown by L.B. Nahmanson that every linearly ordered compact space X, such that Cp (X) is Lindelˆf, is metrizable (see [2]). O.G. Okunev constructed a Tychonoff space X such that X is the union of a countable family of Eberlein compacta and the space Cp (X) is not Lindelˆf [9]. An original description of Eberlein compacta was found by D. Amir and J. Lindenstrauss (see [6, 2, 3]): a compact space X is an Eberlein compactum if and only if there exists a compact subspace F of Cp (X) such that functions in F separate points of X, and F is homeomorphic to the onepoint (Alexandroff) compactiÝcation of some discrete space. From this characterization it follows that a compact space Y is an Eberlein compactum if and only if it can be embedded in a Σ∗ -product of real lines. Recall that the Σ∗ -product of real lines is the set of all points x of R τ (for some τ ) such that, for every ε > 0, the number of coordinates of x not in the interval (−ε, ε) is Ýnite. This ìconcreteî description of the class of Eberlein compacta is instrumental in establishing an ìinnerî criterion for a compact space to be an Eberlein compactum. A family γ of subsets of a space X is called T0 -separating if whenever x and y are distinct points of X, there exists V ∈ γ containg exactly one of the points x and y. Rosenthal proved (see [2, 3]) that a compact space X is an Eberlein compactum if and only if there exists a T0 separating σ -point-Ýnite family of cozero sets in X. It is well known that a compact Hausdorff space X is metrizable if and only if the diagonal ∆X = {(x, x): x ∈ X} is a Gδ -subset of X × X [E]. Clearly, the last condition is equivalent to σ -compactness of the space X × X \ ∆X . G. Gruenhage [5] gave a parallel characterization of Eberlein compacta: a compact Hausdorff space X is an Eberlein compactum if and only if the space X × X \ ∆X is σ -metacompact. An interesting natural question is: when does a Tychonoff space X have a Hausdorff compactiÝcation which is an Eberlein compactum? It was shown in [2] that every metrizable space has such compactiÝcation. On the other hand, not every Moore space has an Eberlein compactiÝcation, since there are separable non-metrizable Moore spaces. With every space X one can associate a sequence of function spaces in pointwise topology. Indeed, put X = Cp,0 (X), and Cp,n (X) = Cp (Cp,n−1 (X)), for positive n ∈ ω. The spaces Cp,n (X) are called iterated function spaces over X. A general impression is that with the growth of n the complexity of the space Cp,n (X) grows. However, if X is a
Section C:
Maps and general types of spaces defined by maps
metrizable compactum, then each Cp,n (X) has a countable network and is, therefore, a hereditarily Lindelˆf, hereditarily separable space. O.V. Sipacheva established a similar, but much deeper, result for Eberlein compacta: if X is an Eberlein compactum, then every iterated function space Cp,n (X) is a Lindelˆf Σ-space [12] (see also [2]). The last statement should be compared to the following fact: if Cp,n (X) is a Kσ δ -space (or a continuous image of such a space), for some n 2, then the space X is Ýnite [2]. References [1] A.V. Arhangelískii, Function spaces in the topology of pointwise convergence, and compact sets, Russian Math. Surveys 39 (5) (1984), 9ñ56. [2] A.V. Arhangelískii, Topological Function Spaces, Math. Appl., Vol. 78, Kluwer Academic, Dordrecht (1992), translated from Russian. [3] A.V. Arhangelískii, Some observations on Cp -theory and bibliography, Topology Appl. 89 (1998), 203ñ221. [4] A. Grothendieck, Criteres de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168ñ186. [5] G. Gruenhage, Covering properties of X2 ⊂ ∆, W -sets and compact subsets of Σ-products, Topology Appl. 17 (1984), 287ñ304. [6] J. Lindenstrauss, Weakly compact sets-their topological properties and the Banach spaces they generate, Ann. Math. Stud., Vol. 69, Princeton Univ. Press, Princeton, NJ (1972), 235ñ273. [7] E.A. Michael and M.E. Rudin, A note on Eberlein compacta, PaciÝc J. Math. 72 (2) (1977), 487ñ495. [8] J. van Mill, A homogeneous Eberlein compact space which is not metrizable, PaciÝc J. Math. 101 (1) (1982), 141ñ146. [9] O.G. Okunev, On Lindelöf Σ-spaces of continuous functions in the pointwise topology, Topology Appl. 49 (1993), 149ñ166. [10] D. Preiss and P. Simon, A weakly pseudocompact subspace of a Banach space is weakly compact, Comment. Math. Univ. Carolin. 15 (4) (1974), 603ñ609. [11] J.D. Pryce, A device of R.J. Whitley’s applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. 23 (3) (1971), 532ñ546. [12] O.V. Sipacheva, The structure of iterated function spaces in the topology of pointwise convergence for Eberlein compacta, Mat. Z. (Math. Notes) 47 (3) (1990), 91ñ99. [13] M. Talagrand, Espaces de Banach faiblement K-analytique, Ann. of Math. 110 (3) (1979), 407ñ438. [14] M. Talagrand, A new countably determined Banach space, Israel J. Math. 47 (1984), 75ñ80.
A.V. Arhangelíski Athens, OH, USA
Eberlein compacta
c-18
145
Eberlein Compacta
Eberlein compacta were introduced to mathematics as compact subspaces of Banach spaces with the weak topology. They have a natural description in terms of spaces of functions with the pointwise topology: a compact space Y is an Eberlein compactum if and only if there exists a compact (Hausdorff) space X such that Y is homeomorphic to a subspace of Cp (X). An equivalent description [2]: a compact space X is an Eberlein compactum if and only if there exists a compact subspace F of Cp (X) such that the functions in F separate the points of X. Yet another characterization [1]: a compact space X is an Eberlein compactum if and only if in Cp (X) there exists a dense σ -compact subspace. The advantage of this criterion is that it describes when a compactum is an Eberlein compactum by a topological property of Cp (X). Arguments involving compact sets of functions play a key role in functional analysis. It turns out that compact parts of Cp (X), where X is compact, have much better convergence properties than Cp (X) itself. In particular, convergence in any Eberlein compactum can be described in terms of usual (countable) convergent sequences, that is, a point x is in the closure of a set A if and only if some sequence (an : n ∈ ω) of points of A converges to x. Such spaces are called Fréchet– Urysohn. On the other hand, for a compact space X, Cp (X) is FrÈchetñUrysohn if and only if the space X is scattered [2]. Since every dyadic compactum of countable tightness is metrizable [E], it follows that every Eberlein compactum which is also a dyadic compactum must be metrizable [2]. Another important property of Eberlein compacta: the closure of any countable subset in an Eberlein compactum is metrizable. Moreover, if A is any subset of an Eberlein compactum X, then the weight of the closure of A in X does not exceed the cardinality of A. Spaces with this property are called monolithic [2]. The importance of monolithicity can be seen from the following simple observation: every separable monolithic compactum is metrizable. Therefore, every separable Eberlein compactum is metrizable. It follows that any Tychonoff or Cantor cube of uncountable weight can serve as an example of a non-Eberlein compactum. In fact, H. Rosenthal proved that every Eberlein compactum with the countable Souslin number is metrizable [2] (see also [3]). This considerably improves the statement on metrizability of separable Eberlein compacta and implies that every non-metrizable compact topological group is a non-Eberlein compactum. Every metrizable compactum is Eberlein, though this is not immediately clear from the deÝnition [2]. The simplest example of a non-metrizable Eberlein compactum is the onepoint compactification of an uncountable discrete space. The class of Eberlein compacta is ìcategoricallyî nice. Indeed, any closed subspace of an Eberlein compactum is an
Eberlein compactum, and the product of any countable family of Eberlein compacta is an Eberlein compactum. We already noted above that the last statement does not expand to uncountable products. Every Hausdorff continuous image of an Eberlein compactum is again an Eberlein compactum. This is a rather deep result, not so easy to prove (see [7]). Monolithicity of Eberlein compacta has another nontrivial corollary. It is known that if a monolithic compact Hausdorff space X is FrÈchetñUrysohn, then X is firstcountable at a dense set of points [2]. It follows that every Eberlein compactum is Ýrst-countable at a dense set of points. Hence, each homogeneous Eberlein compactum X is Ýrst-countable and the cardinality of such X does not exceed 2ω . The Alexandroff double circle is an example of a nonmetrizable Eberlein compactum satisfying the Ýrst axiom of countability at every point. J. van Mill constructed a homogeneous non-metrizable Eberlein compactum [8]. Note that the double circle is the union of two metrizable subspaces (the two individual circles). M.E. Rudin proved that every compactum which is the union of two metrizable subspaces is an Eberlein compactum. However, a compact Hausdorff space which is the union of three metrizable subspaces need not be Eberlein (consider the one-point compactiÝcation of a Ψ -space; the space obtained is not monolithic). Eberlein compacta also enter the picture in connection with the following theorem of Grothendieck: if X is a countably compact space then, for every relatively countably compact subset A of Cp (X), the closure of A in Cp (X) is compact. Note that a subset A of a space Z is said to be relatively countably compact or countably compact in Z if every inÝnite subset of A has a point of accumulation in Z. It turns out that, under these assumptions, the closure of A in Cp (X) is an Eberlein compactum [4, 11, 2]. From the fact that every Eberlein compactum is FrÈchetñUrysohn it follows that every countably compact subset of Cp (X) is compact whenever X is countably compact. However, even more is true: if P is a pseudocompact subspace of Cp (X) where X is countably compact then P is compact and, therefore, closed in Cp (X). A proof of this result is based on the following convergence property of Eberlein compacta: for each point x of any Eberlein compactum X there exists a sequence (Un : n ∈ ω) of non-empty open subsets of X converging to x (D. Preiss and P. Simon [10]). How ìcompactî can Cp (X) be for a compact X? Obviously, Cp (X) is compact only if X is empty. N.V. Velichko showed (see [2]) that Cp (X) is σ -compact if and only if X is Ýnite (then, and only then, Cp (X) is locally compact as well). In this connection the next result is especially interesting: for every Eberlein compactum X, the space Cp (X) is a Kσ δ , that is, Cp (X) can be represented as the intersection of a countable family of σ -compact subspaces of R X
146 [1, 2]. Therefore, for every Eberlein compactum X, Cp (X) is a Lindelöf Σ-space and even a K-analytic space (M. Talagrand [13]). Unfortunately, the Kσ δ property of Cp (X) does not characterize Eberlein compacta; M. Talagrand constructed a compact Hausdorff space X such that Cp (X) is a Kσ δ , and X is not an Eberlein compactum [14]. However, a zero-dimensional compactum X is Eberlein if and only if the space Cp (X, D) of all continuous maps of X in the discrete two-point space D = {0, 1}, in the pointwise topology, is σ -compact [2, 4.6.4]. From Talagrandís theorem (see above) it follows that every linearly ordered compactum which is also an Eberlein compactum must be metrizable. Indeed, it was shown by L.B. Nahmanson that every linearly ordered compact space X, such that Cp (X) is Lindelˆf, is metrizable (see [2]). O.G. Okunev constructed a Tychonoff space X such that X is the union of a countable family of Eberlein compacta and the space Cp (X) is not Lindelˆf [9]. An original description of Eberlein compacta was found by D. Amir and J. Lindenstrauss (see [6, 2, 3]): a compact space X is an Eberlein compactum if and only if there exists a compact subspace F of Cp (X) such that functions in F separate points of X, and F is homeomorphic to the onepoint (Alexandroff) compactiÝcation of some discrete space. From this characterization it follows that a compact space Y is an Eberlein compactum if and only if it can be embedded in a Σ∗ -product of real lines. Recall that the Σ∗ -product of real lines is the set of all points x of R τ (for some τ ) such that, for every ε > 0, the number of coordinates of x not in the interval (−ε, ε) is Ýnite. This ìconcreteî description of the class of Eberlein compacta is instrumental in establishing an ìinnerî criterion for a compact space to be an Eberlein compactum. A family γ of subsets of a space X is called T0 -separating if whenever x and y are distinct points of X, there exists V ∈ γ containg exactly one of the points x and y. Rosenthal proved (see [2, 3]) that a compact space X is an Eberlein compactum if and only if there exists a T0 separating σ -point-Ýnite family of cozero sets in X. It is well known that a compact Hausdorff space X is metrizable if and only if the diagonal ∆X = {(x, x): x ∈ X} is a Gδ -subset of X × X [E]. Clearly, the last condition is equivalent to σ -compactness of the space X × X \ ∆X . G. Gruenhage [5] gave a parallel characterization of Eberlein compacta: a compact Hausdorff space X is an Eberlein compactum if and only if the space X × X \ ∆X is σ -metacompact. An interesting natural question is: when does a Tychonoff space X have a Hausdorff compactiÝcation which is an Eberlein compactum? It was shown in [2] that every metrizable space has such compactiÝcation. On the other hand, not every Moore space has an Eberlein compactiÝcation, since there are separable non-metrizable Moore spaces. With every space X one can associate a sequence of function spaces in pointwise topology. Indeed, put X = Cp,0 (X), and Cp,n (X) = Cp (Cp,n−1 (X)), for positive n ∈ ω. The spaces Cp,n (X) are called iterated function spaces over X. A general impression is that with the growth of n the complexity of the space Cp,n (X) grows. However, if X is a
Section C:
Maps and general types of spaces defined by maps
metrizable compactum, then each Cp,n (X) has a countable network and is, therefore, a hereditarily Lindelˆf, hereditarily separable space. O.V. Sipacheva established a similar, but much deeper, result for Eberlein compacta: if X is an Eberlein compactum, then every iterated function space Cp,n (X) is a Lindelˆf Σ-space [12] (see also [2]). The last statement should be compared to the following fact: if Cp,n (X) is a Kσ δ -space (or a continuous image of such a space), for some n 2, then the space X is Ýnite [2]. References [1] A.V. Arhangelískii, Function spaces in the topology of pointwise convergence, and compact sets, Russian Math. Surveys 39 (5) (1984), 9ñ56. [2] A.V. Arhangelískii, Topological Function Spaces, Math. Appl., Vol. 78, Kluwer Academic, Dordrecht (1992), translated from Russian. [3] A.V. Arhangelískii, Some observations on Cp -theory and bibliography, Topology Appl. 89 (1998), 203ñ221. [4] A. Grothendieck, Criteres de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168ñ186. [5] G. Gruenhage, Covering properties of X2 ⊂ ∆, W -sets and compact subsets of Σ-products, Topology Appl. 17 (1984), 287ñ304. [6] J. Lindenstrauss, Weakly compact sets-their topological properties and the Banach spaces they generate, Ann. Math. Stud., Vol. 69, Princeton Univ. Press, Princeton, NJ (1972), 235ñ273. [7] E.A. Michael and M.E. Rudin, A note on Eberlein compacta, PaciÝc J. Math. 72 (2) (1977), 487ñ495. [8] J. van Mill, A homogeneous Eberlein compact space which is not metrizable, PaciÝc J. Math. 101 (1) (1982), 141ñ146. [9] O.G. Okunev, On Lindelöf Σ-spaces of continuous functions in the pointwise topology, Topology Appl. 49 (1993), 149ñ166. [10] D. Preiss and P. Simon, A weakly pseudocompact subspace of a Banach space is weakly compact, Comment. Math. Univ. Carolin. 15 (4) (1974), 603ñ609. [11] J.D. Pryce, A device of R.J. Whitley’s applied to pointwise compactness in spaces of continuous functions, Proc. London Math. Soc. 23 (3) (1971), 532ñ546. [12] O.V. Sipacheva, The structure of iterated function spaces in the topology of pointwise convergence for Eberlein compacta, Mat. Z. (Math. Notes) 47 (3) (1990), 91ñ99. [13] M. Talagrand, Espaces de Banach faiblement K-analytique, Ann. of Math. 110 (3) (1979), 407ñ438. [14] M. Talagrand, A new countably determined Banach space, Israel J. Math. 47 (1984), 75ñ80.
A.V. Arhangelíski Athens, OH, USA