Cp(omega 1 + 1) is monolithic and Frechet-Urysohn

This is another post that discusses what C_p(X) is like when X is a compact space. In this post, we discuss the example C_p(\omega_1+1) where \omega_1+1 is the first compact uncountable ordinal. Note that \omega_1+1 is the successor to \omega_1, which is the first (or least) uncountable ordinal. The function space C_p(\omega_1+1) is monolithic and is a Frechet-Urysohn space. Interestingly, the first property is possessed by C_p(X) for all compact spaces X. The second property is possessed by all compact scattered spaces. After we discuss C_p(\omega_1+1), we discuss briefly the general results for C_p(X).

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Initial discussion

The function space C_p(\omega_1+1) is a dense subspace of the product space \mathbb{R}^{\omega_1}. In fact, C_p(\omega_1+1) is homeomorphic to a subspace of the following subspace of \mathbb{R}^{\omega_1}:

    \Sigma(\omega_1)=\left\{x \in \mathbb{R}^{\omega_1}: x_\alpha \ne 0 \text{ for at most countably many } \alpha < \omega_1 \right\}

The subspace \Sigma(\omega_1) is the \Sigma-product of \omega_1 many copies of the real line \mathbb{R}. The \Sigma-product of separable metric spaces is monolithic (see here). The \Sigma-product of first countable spaces is Frechet-Urysohn (see here). Thus \Sigma(\omega_1) has both of these properties. Since the properties of monolithicity and being Frechet-Urysohn are carried over to subspaces, the function space C_p(\omega_1+1) has both of these properties. The key to the discussion is then to show that C_p(\omega_1+1) is homeopmophic to a subspace of the \Sigma-product \Sigma(\omega_1).

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Connection to \Sigma-product

We show that the function space C_p(\omega_1+1) is homeomorphic to a subspace of the \Sigma-product of \omega_1 many copies of the real lines. Let Y_0 be the following subspace of C_p(\omega_1+1):

    Y_0=\left\{f \in C_p(\omega_1+1): f(\omega_1)=0 \right\}

Every function in Y_0 has non-zero values at only countably points of \omega_1+1. Thus Y_0 can be regarded as a subspace of the \Sigma-product \Sigma(\omega_1).

By Theorem 1 in this previous post, C_p(\omega_1+1) \cong Y_0 \times \mathbb{R}, i.e, the function space C_p(\omega_1+1) is homeomorphic to the product space Y_0 \times \mathbb{R}. On the other hand, the product Y_0 \times \mathbb{R} can also be regarded as a subspace of the \Sigma-product \Sigma(\omega_1). Basically adding one additional factor of the real line to Y_0 still results in a subspace of the \Sigma-product. Thus we have:

    C_p(\omega_1+1) \cong Y_0 \times \mathbb{R} \subset \Sigma(\omega_1) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

Thus C_p(\omega_1+1) possesses all the hereditary properties of \Sigma(\omega_1). Another observation we can make is that \Sigma(\omega_1) is not hereditarily normal. The function space C_p(\omega_1+1) is not normal (see here). The \Sigma-product \Sigma(\omega_1) is normal (see here). Thus \Sigma(\omega_1) is not hereditarily normal.

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A closer look at C_p(\omega_1+1)

In fact C_p(\omega_1+1) has a stronger property that being monolithic. It is strongly monolithic. We use homeomorphic relation in (1) above to get some insight. Let h be a homeomorphism from C_p(\omega_1+1) onto Y_0 \times \mathbb{R}. For each \alpha<\omega_1, let H_\alpha be defined as follows:

    H_\alpha=\left\{f \in C_p(\omega_1+1): f(\gamma)=0 \ \forall \ \alpha<\gamma<\omega_1 \right\}

Clearly H_\alpha \subset Y_0. Furthermore H_\alpha can be considered as a subspace of \mathbb{R}^\omega and is thus metrizable. Let A be a countable subset of C_p(\omega_1+1). Then h(A) \subset H_\alpha \times \mathbb{R} for some \alpha<\omega_1. The set H_\alpha \times \mathbb{R} is metrizable. The set H_\alpha \times \mathbb{R} is also a closed subset of Y_0 \times \mathbb{R}. Then \overline{A} is contained in H_\alpha \times \mathbb{R} and is therefore metrizable. We have shown that the closure of every countable subspace of C_p(\omega_1+1) is metrizable. In other words, every separable subspace of C_p(\omega_1+1) is metrizable. This property follows from the fact that C_p(\omega_1+1) is strongly monolithic.

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Monolithicity and Frechet-Urysohn property

As indicated at the beginning, the \Sigma-product \Sigma(\omega_1) is monolithic (in fact strongly monolithic; see here) and is a Frechet-Urysohn space (see here). Thus the function space C_p(\omega_1+1) is both strongly monolithic and Frechet-Urysohn.

Let \tau be an infinite cardinal. A space X is \tau-monolithic if for any A \subset X with \lvert A \lvert \le \tau, we have nw(\overline{A}) \le \tau. A space X is monolithic if it is \tau-monolithic for all infinite cardinal \tau. It is straightforward to show that X is monolithic if and only of for every subspace Y of X, the density of Y equals to the network weight of Y, i.e., d(Y)=nw(Y). A longer discussion of the definition of monolithicity is found here.

A space X is strongly \tau-monolithic if for any A \subset X with \lvert A \lvert \le \tau, we have w(\overline{A}) \le \tau. A space X is strongly monolithic if it is strongly \tau-monolithic for all infinite cardinal \tau. It is straightforward to show that X is strongly monolithic if and only if for every subspace Y of X, the density of Y equals to the weight of Y, i.e., d(Y)=w(Y).

In any monolithic space, the density and the network weight coincide for any subspace, and in particular, any subspace that is separable has a countable network. As a result, any separable monolithic space has a countable network. Thus any separable space with no countable network is not monolithic, e.g., the Sorgenfrey line. On the other hand, any space that has a countable network is monolithic.

In any strongly monolithic space, the density and the weight coincide for any subspace, and in particular any separable subspace is metrizable. Thus being separable is an indicator of metrizability among the subspaces of a strongly monolithic space. As a result, any separable strongly monolithic space is metrizable. Any separable space that is not metrizable is not strongly monolithic. Thus any non-metrizable space that has a countable network is an example of a monolithic space that is not strongly monolithic, e.g., the function space C_p([0,1]). It is clear that all metrizable spaces are strongly monolithic.

The function space C_p(\omega_1+1) is not separable. Since it is strongly monolithic, every separable subspace of C_p(\omega_1+1) is metrizable. We can see this by knowing that C_p(\omega_1+1) is a subspace of the \Sigma-product \Sigma(\omega_1), or by using the homeomorphism h as in the previous section.

For any compact space X, C_p(X) is countably tight (see this previous post). In the case of the compact uncountable ordinal \omega_1+1, C_p(\omega_1+1) has the stronger property of being Frechet-Urysohn. A space Y is said to be a Frechet-Urysohn space (also called a Frechet space) if for each y \in Y and for each M \subset Y, if y \in \overline{M}, then there exists a sequence \left\{y_n \in M: n=1,2,3,\cdots \right\} such that the sequence converges to y. As we shall see below, C_p(X) is rarely Frechet-Urysohn.

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General discussion

For any compact space X, C_p(X) is monolithic but does not have to be strongly monolithic. The monolithicity of C_p(X) follows from the following theorem, which is Theorem II.6.8 in [1].

Theorem 1
Then the function space C_p(X) is monolithic if and only if X is a stable space.

See chapter 3 section 6 of [1] for a discussion of stable spaces. We give the definition here. A space X is stable if for any continuous image Y of X, the weak weight of Y, denoted by ww(Y), coincides with the network weight of Y, denoted by nw(Y). In [1], ww(Y) is notated by iw(Y). The cardinal function ww(Y) is the minimum cardinality of all w(T), the weight of T, for which there exists a continuous bijection from Y onto T.

All compact spaces are stable. Let X be compact. For any continuous image Y of X, Y is also compact and ww(Y)=w(Y), since any continuous bijection from Y onto any space T is a homeomorphism. Note that ww(Y) \le nw(Y) \le w(Y) always holds. Thus ww(Y)=w(Y) implies that ww(Y)=nw(Y). Thus we have:

Corollary 2
Let X be a compact space. Then the function space C_p(X) is monolithic.

However, the strong monolithicity of C_p(\omega_1+1) does not hold in general for C_p(X) for compact X. As indicated above, C_p([0,1]) is monolithic but not strongly monolithic. The following theorem is Theorem II.7.9 in [1] and characterizes the strong monolithicity of C_p(X).

Theorem 3
Let X be a space. Then C_p(X) is strongly monolithic if and only if X is simple.

A space X is \tau-simple if whenever Y is a continuous image of X, if the weight of Y \le \tau, then the cardinality of Y \le \tau. A space X is simple if it is \tau-simple for all infinite cardinal numbers \tau. Interestingly, any separable metric space that is uncountable is not \omega-simple. Thus [0,1] is not \omega-simple and C_p([0,1]) is not strongly monolithic, according to Theorem 3.

For compact spaces X, C_p(X) is rarely a Frechet-Urysohn space as evidenced by the following theorem, which is Theorem III.1.2 in [1].

Theorem 4
Let X be a compact space. Then the following conditions are equivalent.

  1. C_p(X) is a Frechet-Urysohn space.
  2. C_p(X) is a k-space.
  3. The compact space X is a scattered space.

A space X is a scattered space if for every non-empty subspace Y of X, there exists an isolated point of Y (relative to the topology of Y). Any space of ordinals is scattered since every non-empty subset has a least element. Thus \omega_1+1 is a scattered space. On the other hand, the unit interval [0,1] with the Euclidean topology is not scattered. According to this theorem, C_p([0,1]) cannot be a Frechet-Urysohn space.

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.

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\copyright \ 2014 \text{ by Dan Ma}

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

Metrization Theorems for Compact Spaces

In this blog I have already presented two metrization theorems for compact spaces: (1) any compact space with a countable network is metrizable (see the post), (2) any compact space with a G_\delta-diagonal is metrizable (see the post). I now present another classic theorem: any countably compact space with a point-countable base is metrizable. This theorem is a classic result of Miscenko ([1]). All spaces are at least Hausdorff and regular. We have the following three metrization theorems for compact spaces. In subsequent posts, I will discuss generalizations of these theorems and discuss related concepts.

Thoerem 1. Any compact space with a countable network is metrizable.
The proof is in this post.

Thoerem 2. Any compact space with a G_\delta-\text{diagonal} is metrizable.
The proof is in this post.

Thoerem 3. Any countably compact space with a point-countable base is metrizable.

A base \mathcal{B} for a space X is a point-countabe base if every point in X belongs to at most countably elements of \mathcal{B}.

Proof of Theorem 3. Let \mathcal{B} be a point-countable base for the countably compact space X. We show that X is separable. Once we have a countable dense subset, the base \mathcal{B} has to be a countable base. So we inductively define a sequence of countable sets \lbrace{D_0,D_1,...}\rbrace such that D=\bigcup_{n<\omega}D_n is dense in X.

Let D_0=\lbrace{x_0}\rbrace be a one-point set to start with. For n>0, let E_n=\bigcup_{i<n}D_n. Let \mathcal{B}_n=\lbrace{B \in \mathcal{B}:B \cap E_n \neq \phi}\rbrace. For each finite T \subset \mathcal{B}_n such that X - \bigcap T \neq \phi, choose a point x(T) \in X - \bigcup T. Let D_n be the union of E_n and the set of all points x(T). Let D=\bigcup_{n<\omega}D_n.

We claim that \overline{D}=X. Suppose we have x \in X-\overline{D}. Let \mathcal{A}=\lbrace{B \in \mathcal{B}:B \cap D \neq \phi \phantom{X} \text{and} \thinspace x \notin B}\rbrace. We know that \mathcal{A} is countable since every element of \mathcal{A} contains points of the countable set D. We also know that \mathcal{A} is an open cover of \overline{D}. By the countably compactness of \overline{D}, we can find a finite T \subset \mathcal{A} such that \overline{D} \subset \bigcup T. The finite set T must have appeared during the induction process of selecting points for D_n for some n (i.e. T \subset \mathcal{B}_n). So a point x(T) has been chosen such that x(T) \notin \bigcup T (thus we have x(T) \in D_n \subset \overline{D}). On the other hand, since \overline{D} \subset \bigcup T, we observe that x(T) \notin \overline{D}, producing a contradiction. Thus the countable set D is dense in X, making the point-countable base \mathcal{B} a countable base.

Reference

  1. Miscenko, A., Spaces with a point-countable base, Dokl. Acad. Nauk SSSR, 144 (1962), 985-988. (English translation: Soviet Math. Dokl. 3 (1962), 1199-1202).

Network Weight of Topological Spaces – II

This is a continuation of the discussion on network. In the previous post, I showed that the network weight (the minimum cardinality of a network) coincides with the weight for both metrizable spaces and locally compact spaces. In another post, I showed that this is true for compact spaces. I now show that this is also true for the class of Moore spaces. First, some definitions. A sequence \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} of open covers of a space X is a development for X if for each x \in X and each open set U \subset X with x \in U, there is some n such that any open set in \mathcal{D}_n containing the point x is contained in U. A developable space is one that has a development. A Moore space is a regular developable space.

For a collection of \mathcal{G} of subsets of a space X and for x \in X, define st(x,\mathcal{G})=\bigcup\lbrace{U \in \mathcal{G}:x \in U}\rbrace. An equivalent way of defining a development: A sequence \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} of open covers of a space X is a development for X if for each x \in X, \lbrace{st(x,\mathcal{G}_n):n \in \omega}\rbrace is a local base at x. For a basic introduction to Moore space and the Moore space conjecture, there are numerous places to look in the literature ([1] being one of them).

Theorem. If X is a Moore space, then nw(X)=w(X).

Proof. Since nw(X) \leq w(X) always holds, we only need to show w(X) \leq nw(X). To this end, we exhibit a base \mathcal{B} with \vert \mathcal{B} \lvert \leq nw(X). Let \lbrace{\mathcal{D}_n}\rbrace_{n<\omega} be a development for X. Let \mathcal{N} be a network with cardinality nw(X).

For each N \in \mathcal{N}, choose open set O(n,N) \in \mathcal{D}_n such that N \subset O(n,N). Let \mathcal{B}_n=\lbrace{O(n,N):N \in \mathcal{N}}\rbrace and \mathcal{B}=\bigcup_{n<\omega}\mathcal{B}_n. Note that \lvert \mathcal{B} \lvert \leq nw(X). Because \mathcal{N} is a network, each \mathcal{B}_n is a cover of X. To see this, let x \in X. Choose some V \in \mathcal{D}_n such that x \in V. There is some N \in \mathcal{N} such that x \in N \subset V. Then x \in O(n,N). For each n, \mathcal{B}_n \subset \mathcal{D}_n. The sequence \lbrace{\mathcal{B}_n}\rbrace works like a development. We have just shown that \mathcal{B} is a base for X.

Corollary. The example of Butterfly space is not a Moore space.

The example of the Butterfly (or Bow-tie) space is defined in this previous post. This space has a countable network and the weight of this space is continuum. Thus this space cannot be a Moore space.

Reference
[1] Steen, L. A. & Seebach, J. A. [1995] Counterexamples in Topology, Dover Books.

Network Weight of Topological Spaces – I

In the previous post, I discussed the notion of network of a topological space. It was noted that for any space X, the network weight (the least cardinality of a network for X) is always \leq the weight (the least cardinality of a base for X). When X is compact, the network weight and weight would coincide. Are there other classes of spaces for which network weight = weight? I would like to discuss two other classes of spaces where network weight and weight coincide, namely metrizable spaces and locally compact spaces. The following two theorems are proved. For a basic discussion on network, see the previous post.

Theorem 1. If X is metrizable, then nw(X)=w(X).

Proof. For the case of w(X)=\omega, we have nw(X)=\omega. Now consider the case that w(X) is an uncountable cardinal. Based on the Bing-Nagata-Smirnov metrization theorem, any metrizable space has a \sigma-discrete base. Let \mathcal{B}=\bigcup_{n<\omega} \mathcal{B}_n be a \sigma-discrete base for the metrizable space X. Now let \mathcal{K}=\lvert \mathcal{B} \lvert. Because each \mathcal{B}_n is a discrete collection of open sets, any network woulld have cardinality at least as big as \lvert \mathcal{B}_n \lvert for each n. If \mathcal{K}=\lvert \mathcal{B}_n \lvert for some n, then \mathcal{K} \leq nw(X). If \mathcal{K} is the least upper bound of \lvert \mathcal{B}_n \lvert, then \mathcal{K} \leq nw(X). Both cases imply w(X) \leq nw(X). Since nw(X) \leq w(X) always hold,  we have w(X)=nw(X).

Theorem 2. If X is a locally compact space, then nw(X)=w(X).

Proof. Let X be locally compact where nw(X)=\mathcal{K}. The idea is that we can obtain a base for X of cardinality \leq \mathcal{K} (i.e. w(X) \leq nw(X)). Let \mathcal{N} be a network whose cardinality is \mathcal{K}. Here’s a sketch of the proof. Each point in X has an open neighborhood whose closure is compact. For the compact closure of such open neighborhood, the weight would coincide with the network weight. Thus we can find a base of size \leq \mathcal{K} within such open neighborhood. Because \lvert \mathcal{N} \lvert=\mathcal{K}, we only need to consider \mathcal{K} many such open neighborhoods with compact closure. Thus we can obtain a base for X of cardinality \leq \mathcal{K}. To make this sketch more precise, consider the following three claims.

Claim 1
The collection of all N \in \mathcal{N}, where \overline{N} is compact, is a cover of the space X.

Claim 2
For every compact set A \subset X, there is an open set U such that A \subset U and \overline{U} is compact.

Claim 3
If U \subset X is open with \overline{U} compact, then we can obtain a base \mathcal{B}_U for the open subspace U with \vert \mathcal{B}_U \vert \leq \mathcal{K}.

For each N \in \mathcal{N} in Claim 1, we can select an open U (as in Claim 2)such that \overline{N} \subset U and \overline{U} is compact. Let \mathcal{B} be the union of all the \mathcal{B}_U in Claim 3 over all such U. There are \leq \mathcal{K} many N in Claim 1. Thus \vert \mathcal{B} \lvert \leq \mathcal{K}. Note that \mathcal{B} would form a base for the whole space X.

Both Claim 1 and Claim 2 are direct consequence of locally compactness. To see Claim 3, let U be open such that \overline{U} is compact. We have nw(\overline{U}) \leq nw(X)=\mathcal{K} (the network weight of a subspace cannot exceed the original network weight). By the result in the previous post, we have w(\overline{U})=nw(\overline{U}). We now have w(U) \leq w(\overline{U}) (the weight of a subspace cannot exceed the weight of the space containing it). So the weight of any open subspace with compact closure cannot exceed \mathcal{K}.

Corollary. For both metrizable spaces and locally compact spaces X, w(X) \leq \lvert X \lvert.

A Short Note About The Sorgenfrey Line

Regarding the Sorgenfrey Line, we have a couple of points to add in addition to the contents in the previous post on the Sorgenfrey line. We show the following:

  • The Sorgenfrey Line does not have a countable network.
  • An alternative proof that S \times S is not normal.

In point G in the previous post, we prove that the Sorgenfrey line has no countable base. So the result in this post improves on the previous post. In point E in the previous post, we prove the Sorgenfrey plane is not normal using the Jones’ Lemma. The alternative method is to use the Baire Category Theorem.

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The first bullet point

Given a space X, given \mathcal{A} a collection of subsets of X, we say \mathcal{A} is a network of X if for each open set U \subset{X} and for each p \in {U}, there is some A \in {\mathcal{A}} such that p \in A. The network weight of X, denoted by nw(X), is the least cardinallity of a network of X.

Of interest here are the spaces with countable network. Note that spaces with countable network are Lindelof. Note that the product of two spaces (each with a countable network) also has a countable network. If S has a countable network, then S \times S would have a countable network and thus Lindelof. So the Sorgenfrey Line has no countable network.

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The second bullet point

To prove that S \times S is not normal using the Baire Category Theorem, define H_0 and H_1 as follows. It can be shown that these two closed subsets of S \times S cannot be separated by disjoint open sets.

    H_0=\lbrace{(x,-x): x} \text{ is rational} \rbrace
    H_1=\lbrace{(y,-y): y} \text{ is irrational} \rbrace

Suppose U_0 and U_1 are open subsets of S \times S such that H_0 \subset{U_0} and H_1 \subset{U_1}. It is shown below that U_0 \cap U_1 \ne \varnothing.

Let \mathbb{P} be the set of all irrational numbers and let \mathbb{Q} be the set of all rational numbers. For each p \in {\mathbb{P}}, choose some real number a(p)>0 such that

    W_p=[p,p+a(p)) \times [-p,-p+a(p)) \subset{U_1}

Let P_n=\lbrace{p \in {\mathbb{P}}: a(p)>\frac{1}{n}}\rbrace. Obviously \mathbb{P}=\bigcup \limits_n {P_n}. Since \mathbb{P} is not an F_\sigma subset of \mathbb{R}, there exists z \in \mathbb{Q} and there exists an n such that z is in the closure of P_n in the usual topology of \mathbb{R}. It is shown below that the point (z,-z) is in the closure of U_1 in S \times S. Since (z,-z) \in H_0 \subset U_0, the open set U_0 would have to contain points of U_1. Thus U_0 \cap U_1 \ne \varnothing.

To see that the point (z,-z) is in the closure of U_1 in S \times S, let V be an open set containing the point (z,-z). To make it easier to work with, assume V is of the form

    V=[z,t) \times [-z,-z+t)

for some positive real number t<\frac{1}{2n}. Since (z,-z) is in the Euclidean closure of P_n, there is a p \in P_n such that \lvert z-p \lvert < \frac{t}{10}. It does not matter whether the point p is to the left or right of z, we have the following two observations:

  • The interval [z,z+t) must overlap with the interval [p,p+t). Then pick x in the intersection.
  • The interval [-z,-z+t) must overlap with the interval [-p,-p+t). Then pick y in the intersection.

Immediately, the point (x,y) belongs to the open set V. Consider the following derivations:

    \displaystyle p < x<p+t<p+\frac{1}{2n}<p+\frac{1}{n}<p+a(p)

    \displaystyle -p < y<-p+t<-p+\frac{1}{2n}<-p+\frac{1}{n}<-p+a(p)

The above derivations show that the point (x,y) belongs to the open set W_p as defined above. The open set W_p is chosen to be a subset of U_1. Thus V \cap U_1 \ne \varnothing, establishing that the point (z,-z) is in the closure of U_1 in S \times S. The proof that the Sorgenfrey plane is not normal is now completed.

Note that in using the Baire Category Theorem, a pair of disjoint closed sets is produced. The proof using the Jones Lemma only implies that such a pair exists.