Sigma-products of separable metric spaces are monolithic

Let \Sigma(\kappa) be the \Sigma-product of \kappa many copies of the real lines where \kappa is any infinite cardinal number. Any compact space that can be embedded in \Sigma(\kappa) for some \kappa is said to be a Corson compact space. Corson compact spaces play an important role in functional analysis. Corson compact spaces are also very interesting from a topological point of view. Some of the properties of Corson compact spaces are inherited (as subspaces) from the \Sigma-product \Sigma(\kappa). One such property is the property that the \Sigma-product \Sigma(\kappa) is monolithic, which implies that the closure of any countable subspace of \Sigma(\kappa) is metrizable.

Previous blog posts on \Sigma-products:

A previous blog post on monolithic spaces: A short note on monolithic spaces. A listing of other blog posts on Corson compact spaces is given at the end of this post.

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Defining Sigma-product

Let \kappa be an infinite cardinal number. For each \alpha<\kappa, let X_\alpha be a topological space. Let b \in \prod_{\alpha<\kappa} X_\alpha. The \Sigma-product of the spaces X_\alpha about the base point b is defined as follows:

    \Sigma_{\alpha<\kappa} X_\alpha=\left\{x \in \prod_{\alpha<\kappa} X_\alpha: x_\alpha \ne b_\alpha \text{ for at most countably many } \alpha < \kappa \right\}

If each X_\alpha=\mathbb{R} and if the base point b is such that b_\alpha=0 for all \alpha<\kappa, then we use the notation \Sigma(\kappa) for \Sigma_{\alpha<\kappa} X_\alpha, i.e., \Sigma(\kappa) is defined as follows:

    \Sigma(\kappa)=\left\{x \in \mathbb{R}^\kappa: x_\alpha \ne 0 \text{ for at most countably many } \alpha < \kappa \right\}

A compact space is said to be a Corson compact space if it can be embedded in the \Sigma-product \Sigma(\kappa) for some infinite cardinal \kappa.

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Monolithic Spaces

A space X is monolithic if for every subspace Y of X, the density of Y equals the network weight of Y, i.e., d(Y)=nw(Y). A space X is strongly monolithic if for every subspace Y of X, the density of Y equals the weight of Y, i.e., d(Y)=w(Y). See the previous post called A short note on monolithic spaces.

The proof of the fact that \Sigma-product of separable metrizable spaces is monolithic can be worked out quite easily from definitions. Interested readers are invited to walk through the proof. For the sake of completeness, we prove the following theorem.

Theorem 1
Suppose that for each \alpha<\kappa, X_\alpha is a separable metric space. Then the \Sigma-product \Sigma_{\alpha<\kappa} X_\alpha is strongly monolithic.

Proof of Theorem 1
Let b be the base point of the \Sigma-product X=\Sigma_{\alpha<\kappa} X_\alpha. For each x \in X, let S(x) be the support of the point x, i.e., the set of all \alpha<\kappa such that x_\alpha \ne b_\alpha. Let Y be a subspace of X. We show that d(Y)=w(Y).

Let T=\left\{t_\delta: \delta<\tau \right\} be a dense subspace of Y such that d(Y)=\lvert T \lvert=\tau. Note that \overline{T}=Y (closure is taken in Y). Let S=\bigcup_{\delta<\tau} S(t_\delta). Clearly \lvert S \lvert \le \tau. Consider the following subspace of X:

    X(S)=\left\{x \in X: S(x) \subset S  \right\}

It is clear that X(S) is a closed subspace of X. Since T \subset X(S), the closure of T (closure in X or in Y) is a subspace of X(S). Thus Y \subset X(S). Note that \overline{T}=Y \subset X(S). Since each X_\alpha has a countable base, the product space \prod_{\alpha<\tau} X_\alpha has a base of cardinality \tau. Thus \prod_{\alpha<\tau} X_\alpha has weight \le \tau. Since X(S) \subset \prod_{\alpha<\tau} X_\alpha, both Y and X(S) have weights \le \tau. We have w(Y) \le d(Y)=\tau. Note that d(Y) \le w(Y) always holds. Therefore d(Y)=w(Y). \blacksquare

Corollary 2
For any infinite cardinal \kappa, the \Sigma-product \Sigma(\kappa) is strongly monolithic.

Corollary 3
Any Corson compact space is strongly monolithic.

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Blog posts on Corson compact spaces

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\copyright \ 2014 \text{ by Dan Ma}

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A short note on monolithic spaces

In a metrizable space, the density, the network weight and the weight (and several other cardinal functions) always agree (see Theorem 4.1.15 in [2]). This is not the case for topological spaces in general. One handy example is the Sorgenfrey line where the density is \omega (the Sorgenfrey line is separable) and the network weight is continuum (the cardinality of real line). In a monolithic space, the density character and the network weight for any subspace always coincide. Thus metrizable spaces are monolithic. One interesting example of a monolithic space is the \Sigma-product of real lines. A compact space is said to be a Corson compact space if it can be embedded in a \Sigma-product of real lines. Thus Corson compact spaces are monolithic spaces. As a result, any separable subspace of a Corson compact space is metrizable. On the other hand, any separable non-metrizable compact space cannot be Corson compact. This is an introductory discussion of monolithic spaces and is the first post in a series of posts on Corson compact spaces. A listing of other blog posts on Corson compact spaces is given at the end of this post.

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Density and Network Weight

For any set A, the symbol \lvert A \lvert denotes the cardinality of the set A. For any space X, the density of X, denoted by d(X) is the minimum cardinality of a dense subset, i.e., d(X) is the least cardinal number \kappa such that if Y is dense subset of X, then \kappa \le \lvert Y \lvert. If X is separable, then d(X)=\omega.

For any space X, a family \mathcal{N} of subsets of X is a network in the space X if for any x \in X and for any open subset U of X with x \in U, there exists some J \in \mathcal{N} such that x \in J \subset U. In other words, any non-empty open subset of X is the union of elements of the network \mathcal{N}. The network weight of X, denoted by nw(X), is the minimum cardinality of a network in the space X, i.e., nw(X) is the least cardinal number \kappa such that if \mathcal{N} is a network for the space X, then \kappa \le \lvert \mathcal{N} \lvert.

For any space X, the weight of X, denoted by w(X), is the minimum cardinality of a base for the space X, i.e., w(X) is the least cardinal number \kappa such that if \mathcal{B} is a base for the space X, then \kappa \le \lvert \mathcal{B} \lvert. If w(X)=\omega, then X is a space with a countable base (it is a separable metric space). If nw(X)=\omega, X is a space with a countable network. Having a countable network is a strong property, it implies that the space is hereditarily Lindelof (hence hereditarily normal) and hereditarily separable (see this previous post). However, having a countable network is not as strong as having a countable base. The function space C_p(\mathbb{R}) has a countable network (see this previous post) and fails to be first countable at every point.

If \mathcal{N} is a network for the space X, then picking a point from each set in \mathcal{N} will produce a dense subset of X. Then d(X) \le nw(X) for any space X. In general nw(X) \le d(X) does not hold, as indicated by the Sorgenfrey line. Monolithic spaces form a class of spaces in which the inequality nw \le d holds for each space in the class and for each subspace of such a space.

Likewise, the inequality d(X) \le w(X) always holds. The inequality w(X) \le d(X) only holds for a restricted class of spaces. On the other hand, it is clear that nw(X) \le w(X) for any space X.

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Monolithic Spaces

Let \tau be an infinite cardinal number. A space X is said to be \tau-monolithic if for each subspace Y of X with \lvert Y \lvert \le \tau, nw(\overline{Y}) \le \tau. It is easy to verify that the following two statements are equivalent:

  1. X is \tau-monolithic for each infinite cardinal number \tau.
  2. For each subspace Y of X, d(Y)=nw(Y).

A space X is monolithic if either statement 1 or statement 2 holds. In a \omega-monolithic space, any separable subspace has a countable network.

A space X is said to be strongly \tau-monolithic if for each subspace Y of X with \lvert Y \lvert \le \tau, w(\overline{Y}) \le \tau. It is easy to verify that the following two statements are equivalent:

  1. X is strongly \tau-monolithic for each infinite cardinal number \tau.
  2. For each subspace Y of X, d(Y)=w(Y).

A space X is strongly monolithic if either statement 3 or statement 4 holds. In a strongly \omega-monolithic space, any separable subspace is metrizable. It is clear that any strongly monolithic space is monolithic. As indicated below, C_p(\mathbb{R}) is an example of a monolithic space that is not strongly monolithic. However, the two notions coincide for compact spaces. Note that for any compact space, the weight and network weight coincide. Thus if a compact space is monolithic, it is strongly monolithic.

It is also clear that the property of being monolithic is hereditary. Monolithicity is a notion used in C_p-theory and the study of Corson compact spaces (see [1]).

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Examples

Some examples of monolithic spaces are:

  • Metrizable spaces.
  • Any space with a countable network.
  • \Sigma-product of separable metric spaces.
  • The space \omega_1 of countable ordinals.

In fact, with the exception of the spaces with countable networks, the above examples are strongly monolithic. It is well known that the density and weight always agree for metrizable space. \Sigma-product of separable metric spaces is strongly monolithic (shown in this subsequent post). In the space \omega_1, any countable subset is separable and metrizable and any uncountable subset has both density and weight =\omega_1.

If X is a space with a countable network, then for any subspace Y, d(Y)=nw(Y)=\omega. Thus any space with a countable network is monolithic. However, any space that has a countable network but is not metrizable is not strongly monolithic, e.g., the function space C_p(\mathbb{R}). The following proposition about compact monolithic spaces is useful.

Proposition 1
Let X be a compact and monolithic space. Then X is metrizable if and only if X is separable.

Proof of Proposition 1
For the \Longrightarrow direction, note that any compact metrizable space is separable (monolithicity is not needed). For the \Longleftarrow direction, note that any separable monolithic space has a countable network. Any compact space with a countable network is metrizable (see here). \blacksquare

Now consider some spaces that are not monolithic. As indicated above, any space in which the density does not agree with the network weight (in the space or in a subspace) is not monolithic. Proposition 1 indicates that any separable non-metrizable compact space is not monolithic. Examples include the Alexandroff double arrow space ( see here) and the product space I^{\omega_1} where I is the closed unit interval [0,1] with the usual Euclidean topology.

Interestingly, “compact” in Proposition 1 can be replaced by pseudocompact because of the following:

Proposition 2
Let X be a separable pseudocompact and monolithic space. Then X is compact.

Proof of Proposition 2
Any separable monolithic space has a countable network. Any space with a countable network is Lindelof (and hence metacompact). Any pseudocompact metacompact space is compact (see here). \blacksquare

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Blog posts on Corson compact spaces

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  2. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.

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\copyright \ 2014 \text{ by Dan Ma}

Stone-Cech Compactification is Maximal

Let X be a completely regular space. Let \beta X be the Stone-Cech compactification of X. In a previous post, we show that among all compactifcations of X, the Stone-Cech compactification \beta X is maximal with respect to a partial order \le (see Theorem C2 in Two Characterizations of Stone-Cech Compactification). As a result of the maximality, \beta X is the largest among all compactifications of X both in terms of cardinality and weight. We also establish an upper bound for the cardinality of \beta X and an upper bound for the weight of \beta X. As a result, we have upper bounds for cardinalities and weights for all compactifications of X. We prove the following points.

    Upper Bounds for Stone-Cech Compactification

  1. \lvert \beta X \lvert \le 2^{2^{d(X)}}.
  2. w(\beta X) \le 2^{d(X)}.
  3. Stone-Cech Compactification is Maximal

  4. For every compactification \alpha X of the space X, \lvert \alpha X \lvert \le \lvert \beta X \lvert.
  5. For every compactification \alpha X of the space X, w(\alpha X) \le w(\beta X).
  6. Upper Bounds for all Compactifications

  7. For every compactification \alpha X of the space X, w(\alpha X) \le 2^{d(X)}.
  8. For every compactification \alpha X of the space X, \lvert \alpha X \lvert \le 2^{2^{d(X)}}.

It is clear that Results 5 and 6 follow from the preceding results. The links for other posts on Stone-Cech compactification can be found toward the end of this post

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Some Cardinal Functions

Let X be a space. The density of X is denoted by d(X) and is defined to be the smallest cardinality of a dense set in X. For example, if X is separable, then d(X)=\omega. The weight of the space X is denoted by w(X) and is defined to be the smallest cardinality of a base of the space X. For example, if X is second countable (i.e. having a countable space), then w(X)=\omega. Both d(X) and w(X) are cardinal functions that are commonly used in topological discussion. Most authors require that cardinal functions only take on infinite cardinals. We also adopt this convention here. We use c to denote the cardinality of the continuum (the cardinality of the real line \mathbb{R}).

If \mathcal{K} is a cardinal number, then 2^{\mathcal{K}} refers to the cardinal number that is the cardinallity of the set of all functions from \mathcal{K} to 2=\left\{0,1 \right\}. Equivalently, 2^{\mathcal{K}} is also the cardinality of the power set of \mathcal{K} (i.e. the set of all subsets of \mathcal{K}). If \mathcal{K}=\omega (the first infinite ordinal), then 2^\omega=c is the cardinality of the continuum.

If X is separable, then d(X)=\omega (as noted above) and we have 2^{d(X)}=c and 2^{2^{d(X)}}=2^c. Result 5 and Result 6 imply that 2^c is an upper bound for the cardinality of all compactifications of any separable space X and c is an upper bound of the weight of all compactifications of any separable space X.

In general, Result 5 and Result 6 indicate that the density of X bounds the cardinality of any compactification of X by two exponents and the density of X bounds the weight of any compactification of X by one exponent.

Another cardinal function related to weight is that of the network weight. A collection \mathcal{N} of subsets of the space X is said to be a network for X if for each point x \in X and for each open subset U of X with x \in U, there is some set A \in \mathcal{N} with x \in A \subset U. Note that sets in a network do not have to be open. However, any base for a topology is a network. The network weight of the space X is denoted by nw(X) and is defined to be the least cardinality of a network for X. Since any base is a network, we have nw(X) \le w(X). It is also clear that nw(X) \le \lvert X \lvert for any space X. Our interest in network and network weight is to facilitate the discussion of Lemma 2 below. It is a well known fact that in a compact space, the weight and the network weight are the same (see Result 5 in Spaces With Countable Network).
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Some Basic Facts

We need the following two basic results.

    Lemma 1
    Let X be a space. Let C(X) be the set of all continuous functions f:X \rightarrow \mathbb{R}. Then \lvert C(X) \lvert \le 2^{d(X)}.

    Lemma 2
    Let S be a space and let T be a compact space. Suppose that T is the continuous image of S. Then w(T) \le w(S).

Proof of Lemma 1
Let A \subset X be a dense set with \lvert A \lvert=2^{d(X)}. Let \mathbb{R}^A be the set of all functions from A to \mathbb{R}. Consider the map W:C(X) \rightarrow \mathbb{R}^A by W(f)= f \upharpoonright A. This is a one-to-one map since f=g whenever f and g agree on a dense set. Thus we have \lvert C(X) \lvert \le \lvert \mathbb{R}^A \lvert. Upon doing some cardinal arithmetic, we have \lvert \mathbb{R}^A \lvert=2^{d(X)}. Thus Lemma 1 is established. \blacksquare

Proof of Lemma 2
Let g:S \rightarrow T be a continuous function from S onto T. Let \mathcal{B} be a base for S such that \lvert \mathcal{B} \lvert=w(S). Let \mathcal{N} be the set of all g(B) where B \in \mathcal{B}. Note that \mathcal{N} is a network for T (since g is a continuous function). So we have nw(T) \le \lvert \mathcal{N} \lvert \le \lvert \mathcal{B} \lvert = w(S). Since T is compact, w(T)=nw(T) (see Result 5 in Spaces With Countable Network). Thus we have nw(T)=w(T) \lvert \le w(S). \blacksquare

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Results 1 and 2

Let X be a completely regular space. Let I be the unit interval [0,1]. We show that the Stone-Cech compactification \beta X can be regarded as a subspace of the product space I^{\mathcal{K}} where \mathcal{K}= 2^{d(X)} (the product of 2^{d(X)} many copies of I). The cardinality of I^{\mathcal{K}} is 2^{2^{d(X)}}, thus leading to Result 1.

Let C(X,I) be the set of all continuous functions f:X \rightarrow I. The Stone-Cech compactification \beta X is constructed by embedding X into the product space \prod \limits_{f \in C(X,I)} I_f where each I_f=I (see Embedding Completely Regular Spaces into a Cube or A Beginning Look at Stone-Cech Compactification). Thus \beta X is a subspace of I^{\mathcal{K}_1} where \mathcal{K}_1=\lvert C(X,I) \lvert.

Note that C(X,I) \subset C(X). Thus \beta X can be regarded as a subspace of I^{\mathcal{K}_2} where \mathcal{K}_2=\lvert C(X) \lvert. By Lemma 1, \beta X can be regarded as a subspace of the product space I^{\mathcal{K}} where \mathcal{K}= 2^{d(X)}.

To see Result 2, note that the weight of I^{\mathcal{K}} where \mathcal{K}= 2^{d(X)} is 2^{d(X)}. Then \beta X, as a subspace of the product space, must have weight \le 2^{d(X)}. \blacksquare

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Results 3 and 4

What drives Result 3 and Result 4 is the following theorem (established in Two Characterizations of Stone-Cech Compactification).

    Theorem C2
    Let X be a completely regular space. Among all compactifications of the space X, the Stone-Cech compactification \beta X of the space X is maximal with respect to the partial order \le.

    \text{ }

To define the partial order, for \alpha_1 X and \alpha_2 X, both compactifications of X, we say that \alpha_2 X \le \alpha_1 X if there is a continuous function f:\alpha_1 X \rightarrow \alpha_2 X such that f \circ \alpha_1=\alpha_2. See the following figure.

Figure 1

In this post, we use \le to denote this partial order as well as the order for cardinal numbers. Thus we need to rely on context to distinguish this partial order from the order for cardinal numbers.

Let \alpha X be a compactification of X. Theorem C2 indicates that \alpha X \le \beta X (partial order), which means that there is a continuous f:\beta X \rightarrow \alpha X such that f \circ \beta=\alpha (the same point in X is mapped to itself by f). Note that \alpha X is the image of \beta X under the function f:\beta X \rightarrow \alpha X. Thus we have \lvert \alpha X \lvert \le \lvert \beta X \lvert (cardinal number order). Thus Result 3 is established.

By Lemma 2, the existence of the continuous function f:\beta X \rightarrow \alpha X implies that w(\alpha X) \le w(\beta X) (cardinal number order). Thus Result 4 is established.

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Blog Posts on Stone-Cech Compactification

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Reference

  1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
  2. Willard, S., General Topology, Addison-Wesley Publishing Company, 1970.

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\copyright \ \ 2012