Looking for spaces in which every compact subspace is metrizable

Once it is known that a topological space is not metrizable, it is natural to ask, from a metrizability standpoint, which subspaces are metrizable, e.g. whether every compact subspace is metrizable. This post discusses several classes of spaces in which every compact subspace is metrizable. Though the goal here is not to find a complete characterization of such spaces, this post discusses several classes of spaces and various examples that have this property. The effort brings together many interesting basic and well known facts. Thus the notion “every compact subspace is metrizable” is an excellent learning opportunity.

Several Classes of Spaces

The notion “every compact subspace is metrizable” is a very broad class of spaces. It includes well known spaces such as Sorgenfrey line, Michael line and the first uncountable ordinal \omega_1 (with the order topology) as well as Moore spaces. Certain function spaces are in the class “every compact subspace is metrizable”. The following diagram is a good organizing framework.

    \displaystyle \begin{aligned} &1. \ \text{Metrizable} \\&\ \ \ \ \ \ \ \ \ \Downarrow \\&2. \ \text{Submetrizable} \Longleftarrow 5. \ \exists \ \text{countable network} \\&\ \ \ \ \ \ \ \ \ \Downarrow \\&3. \ \exists \ G_\delta \text{ diagonal} \\&\ \ \ \ \ \ \ \ \ \Downarrow \\&4. \ \text{Every compact subspace is metrizable}  \end{aligned}

Let (X, \tau) be a space. It is submetrizable if there is a topology \tau_1 on the set X such that \tau_1 \subset \tau and (X, \tau_1) is a metrizable space. The topology \tau_1 is said to be weaker (coarser) than \tau. Thus a space X is submetrizable if it has a weaker metrizable topology.

Let \mathcal{N} be a set of subsets of the space X. \mathcal{N} is said to be a network for X if for every open subset O of X and for each x \in O, there exists N \in \mathcal{N} such that x \in N \subset O. Having a network that is countable in size is a strong property (see here for a discussion on spaces with a countable network).

The diagonal of the space X is the subset \Delta=\left\{(x,x): x \in X \right\} of the square X \times X. The space X has a G_\delta-diagonal if \Delta is a G_\delta-subset of X \times X, i.e. \Delta is the intersection of countably many open subsets of X \times X.

The implication 1 \Longrightarrow 2 is clear. For 5 \Longrightarrow 2, see Lemma 1 in this previous post on countable network. The implication 2 \Longrightarrow 3 is left as an exercise. To see 3 \Longrightarrow 4, let K be a compact subset of X. The property of having a G_\delta-diagonal is hereditary. Thus K has a G_\delta-diagonal. According to a well known result, any compact space with a G_\delta-diagonal is metrizable (see here).

None of the implications in the diagram is reversible. The first uncountable ordinal \omega_1 is an example for 4 \not \Longrightarrow 3. This follows from the well known result that any countably compact space with a G_\delta-diagonal is metrizable (see here). The Mrowka space is an example for 3 \not \Longrightarrow 2 (see here). The Sorgenfrey line is an example for both 2 \not \Longrightarrow 5 and 2 \not \Longrightarrow 1.

To see where the examples mentioned earlier are placed, note that Sorgenfrey line and Michael line are submetrizable, both are submetrizable by the usual Euclidean topology on the real line. Each compact subspace of the space \omega_1 is countable and is thus contained in some initial segment [0,\alpha] which is metrizable. Any Moore space has a G_\delta-diagonal. Thus compact subspaces of a Moore space are metrizable.

Function Spaces

We now look at some function spaces that are in the class “every compact subspace is metrizable.” For any Tychonoff space (completely regular space) X, C_p(X) is the space of all continuous functions from X into \mathbb{R} with the pointwise convergence topology (see here for basic information on pointwise convergence topology).

Theorem 1
Suppose that X is a separable space. Then every compact subspace of C_p(X) is metrizable.

Proof
The proof here actually shows more than is stated in the theorem. We show that C_p(X) is submetrizable by a separable metric topology. Let Y be a countable dense subspace of X. Then C_p(Y) is metrizable and separable since it is a subspace of the separable metric space \mathbb{R}^{\omega}. Thus C_p(Y) has a countable base. Let \mathcal{E} be a countable base for C_p(Y).

Let \pi:C_p(X) \longrightarrow C_p(Y) be the restriction map, i.e. for each f \in C_p(X), \pi(f)=f \upharpoonright Y. Since \pi is a projection map, it is continuous and one-to-one and it maps C_p(X) onto C_p(Y). Thus \pi is a continuous bijection from C_p(X) onto C_p(Y). Let \mathcal{B}=\left\{\pi^{-1}(E): E \in \mathcal{E} \right\}.

We claim that \mathcal{B} is a base for a topology on C_p(X). Once this is established, the proof of the theorem is completed. Note that \mathcal{B} is countable and elements of \mathcal{B} are open subsets of C_p(X). Thus the topology generated by \mathcal{B} is coarser than the original topology of C_p(X).

For \mathcal{B} to be a base, two conditions must be satisfied – \mathcal{B} is a cover of C_p(X) and for B_1,B_2 \in \mathcal{B}, and for f \in B_1 \cap B_2, there exists B_3 \in \mathcal{B} such that f \in B_3 \subset B_1 \cap B_2. Since \mathcal{E} is a base for C_p(Y) and since elements of \mathcal{B} are preimages of elements of \mathcal{E} under the map \pi, it is straightforward to verify these two points. \square

Theorem 1 is actually a special case of a duality result in C_p function space theory. More about this point later. First, consider a corollary of Theorem 1.

Corollary 2
Let X=\prod_{\alpha<c} X_\alpha where c is the cardinality continuum and each X_\alpha is a separable space. Then every compact subspace of C_p(X) is metrizable.

The key fact for Corollary 2 is that the product of continuum many separable spaces is separable (this fact is discussed here). Theorem 1 is actually a special case of a deep result.

Theorem 3
Suppose that X=\prod_{\alpha<\kappa} X_\alpha is a product of separable spaces where \kappa is any infinite cardinal. Then every compact subspace of C_p(X) is metrizable.

Theorem 3 is a much more general result. The product of any arbitrary number of separable spaces is not separable if the number of factors is greater than continuum. So the proof for Theorem 1 will not work in the general case. This result is Problem 307 in [2].

A Duality Result

Theorem 1 is stated in a way that gives the right information for the purpose at hand. A more correct statement of Theorem 1 is: X is separable if and only if C_p(X) is submetrizable by a separable metric topology. Of course, the result in the literature is based on density and weak weight.

The cardinal function of density is the least cardinality of a dense subspace. For any space Y, the weight of Y, denoted by w(Y), is the least cardinaility of a base of Y. The weak weight of a space X is the least w(Y) over all space Y for which there is a continuous bijection from X onto Y. Thus if the weak weight of X is \omega, then there is a continuous bijection from X onto some separable metric space, hence X has a weaker separable metric topology.

There is a duality result between density and weak weight for X and C_p(X). The duality result:

The density of X coincides with the weak weight of C_p(X) and the weak weight of X coincides with the density of C_p(X). These are elementary results in C_p-theory. See Theorem I.1.4 and Theorem I.1.5 in [1].

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  2. Tkachuk V. V., A C_p-Theory Problem Book, Topological and Function Spaces, Springer, New York, 2011.

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\copyright 2017 – Dan Ma

Comparing two function spaces

Let \omega_1 be the first uncountable ordinal, and let \omega_1+1 be the successor ordinal to \omega_1. Furthermore consider these ordinals as topological spaces endowed with the order topology. It is a well known fact that any continuous real-valued function f defined on either \omega_1 or \omega_1+1 is eventually constant, i.e., there exists some \alpha<\omega_1 such that the function f is constant on the ordinals beyond \alpha. Now consider the function spaces C_p(\omega_1) and C_p(\omega_1+1). Thus individually, elements of these two function spaces appear identical. Any f \in C_p(\omega_1) matches a function f^* \in C_p(\omega_1+1) where f^* is the result of adding the point (\omega_1,a) to f where a is the eventual constant real value of f. This fact may give the impression that the function spaces C_p(\omega_1) and C_p(\omega_1+1) are identical topologically. The goal in this post is to demonstrate that this is not the case. We compare the two function spaces with respect to some convergence properties (countably tightness and Frechet-Urysohn property) as well as normality.

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Tightness

One topological property that is different between C_p(\omega_1) and C_p(\omega_1+1) is that of tightness. The function space C_p(\omega_1+1) is countably tight, while C_p(\omega_1) is not countably tight.

Let X be a space. The tightness of X, denoted by t(X), is the least infinite cardinal \kappa such that for any A \subset X and for any x \in X with x \in \overline{A}, there exists B \subset A for which \lvert B \lvert \le \kappa and x \in \overline{B}. When t(X)=\omega, we say that X has countable tightness or is countably tight. When t(X)>\omega, we say that X has uncountable tightness or is uncountably tight.

First, we show that the tightness of C_p(\omega_1) is greater than \omega. For each \alpha<\omega_1, define f_\alpha: \omega_1 \rightarrow \left\{0,1 \right\} such that f_\alpha(\beta)=0 for all \beta \le \alpha and f_\alpha(\beta)=1 for all \beta>\alpha. Let g \in C_p(\omega_1) be the function that is identically zero. Then g \in \overline{F} where F is defined by F=\left\{f_\alpha: \alpha<\omega_1 \right\}. It is clear that for any countable B \subset F, g \notin \overline{B}. Thus C_p(\omega_1) cannot be countably tight.

The space \omega_1+1 is a compact space. The fact that C_p(\omega_1+1) is countably tight follows from the following theorem.

Theorem 1
Let X be a completely regular space. Then the function space C_p(X) is countably tight if and only if X^n is Lindelof for each n=1,2,3,\cdots.

Theorem 1 is a special case of Theorem I.4.1 on page 33 of [1] (the countable case). One direction of Theorem 1 is proved in this previous post, the direction that will give us the desired result for C_p(\omega_1+1).

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The Frechet-Urysohn property

In fact, C_p(\omega_1+1) has a property that is stronger than countable tightness. The function space C_p(\omega_1+1) is a Frechet-Urysohn space (see this previous post). Of course, C_p(\omega_1) not being countably tight means that it is not a Frechet-Urysohn space.

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Normality

The function space C_p(\omega_1+1) is not normal. If C_p(\omega_1+1) is normal, then C_p(\omega_1+1) would have countable extent. However, there exists an uncountable closed and discrete subset of C_p(\omega_1+1) (see this previous post). On the other hand, C_p(\omega_1) is Lindelof. The fact that C_p(\omega_1) is Lindelof is highly non-trivial and follows from [2]. The author in [2] showed that if X is a space consisting of ordinals such that X is first countable and countably compact, then C_p(X) is Lindelof.

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Embedding one function space into the other

The two function space C_p(\omega_1+1) and C_p(\omega_1) are very different topologically. However, one of them can be embedded into the other one. The space \omega_1+1 is the continuous image of \omega_1. Let g: \omega_1 \longrightarrow \omega_1+1 be a continuous surjection. Define a map \psi: C_p(\omega_1+1) \longrightarrow C_p(\omega_1) by letting \psi(f)=f \circ g. It is shown in this previous post that \psi is a homeomorphism. Thus C_p(\omega_1+1) is homeomorphic to the image \psi(C_p(\omega_1+1)) in C_p(\omega_1). The map g is also defined in this previous post.

The homeomposhism \psi tells us that the function space C_p(\omega_1), though Lindelof, is not hereditarily normal.

On the other hand, the function space C_p(\omega_1) cannot be embedded in C_p(\omega_1+1). Note that C_p(\omega_1+1) is countably tight, which is a hereditary property.

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Remark

There is a mapping that is alluded to at the beginning of the post. Each f \in C_p(\omega_1) is associated with f^* \in C_p(\omega_1+1) which is obtained by appending the point (\omega_1,a) to f where a is the eventual constant real value of f. It may be tempting to think of the mapping f \rightarrow f^* as a candidate for a homeomorphism between the two function spaces. The discussion in this post shows that this particular map is not a homeomorphism. In fact, no other one-to-one map from one of these function spaces onto the other function space can be a homeomorphism.

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Reference

  1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
  2. Buzyakova, R. Z., In search of Lindelof C_p‘s, Comment. Math. Univ. Carolinae, 45 (1), 145-151, 2004.

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\copyright \ 2014 \text{ by Dan Ma}

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 useful representation of Cp(X)

Let X be a completely regular space. The space C_p(X) is the space of all real-valued continuous functions defined on X endowed with the pointwise convergence topology. In this post, we show that C_p(X) can be represented as the product of a subspace of C_p(X) with the real line \mathbb{R}. We prove the following theorem. See here for an application of this theorem.

Theorem 1
Let X be a completely regular space. Let x \in X. Let Y be defined by:

    Y=\left\{f \in C_p(X): f(x)=0 \right\}

Then C_p(X) is homeomorphic to Y \times \mathbb{R}.

The above theorem can be found in [1] (see Theorem I.5.4 on p. 37). In [1], the homeomorphism is stated without proof. For the sake of completeness, we provide a detailed proof of Theorem 1.

Proof of Theorem 1
Define h: C_p(X) \rightarrow Y \times \mathbb{R} by h(f)=(f-f(x),f(x)) for any f \in C_p(X). The map h is a homeomorphism.

The map is one-to-one

First, we show that it is a one-to-one map. Let f,g \in C_p(X) where f \ne g. Assume that f(x) \ne g(x). Then h(f) \ne h(g). So assume that f(x)=g(x). Then the functions f-f(x) and g-g(x) are different, which means h(f) \ne h(g).

The map is onto

Now we show h maps C_p(X) onto Y \times \mathbb{R}. Let (g,t) \in Y \times \mathbb{R}. Let f=g+t. Note that f(x)=g(x)+t=t. Then f-f(x)=g. We have h(f)=(g,t).

Note. Showing the continuity of h and h^{-1} is a matter of working with the basic open sets in the function space carefully (e.g. making the necessary shifting). Some authors just skip the details and declare them continuous, e.g. [1]. Readers are welcome to work out enough of the details to see the key idea.

The map is continuous

Show that h is continuous. Let f \in C_p(X). Let U \times V be an open set in Y \times \mathbb{R} such that h(f) \in U \times V and,

    U=\left\{g \in Y: \forall \ i=1,\cdots,n, g(x_i) \in U_i \right\}

    \forall \ i=1,\cdots,n, \  U_i=(f(x_i)-f(x)-\frac{1}{k},f(x_i)-f(x)+\frac{1}{k})

    V=(f(x)-\frac{1}{k},f(x)+\frac{1}{k})

where x_1,\cdots,x_n are arbitrary points in X and k is some large positive integer. Define the following:

    \forall \ i=1,\cdots,n, \ W_i=(f(x_i)-\frac{1}{2k},f(x_i)+\frac{1}{2k})

    W_{n+1}=(f(x)-\frac{1}{2k},f(x)+\frac{1}{2k})

    x_{n+1}=x

Then define the open set W as follows:

    W=\left\{q \in C_p(X): \forall \ i=1,\cdots,n,n+1, q(x_i) \in W_i \right\}

Clearly f \in W. We need to show h(W) \subset U \times V. Let q \in W. Then h(q)=(q-q(x),q(x)). We need to show that q-q(x) \in U and q(x) \in V. Note that q(x_{n+1})=q(x) \in W_{n+1}. For each i=1,\cdots,n, q(x_i) \in W_i. So we have the following:

    f(x_i)-\frac{1}{2k}<q(x_i)<f(x_i)+\frac{1}{2k}

    f(x)-\frac{1}{2k}<q(x)<f(x)+\frac{1}{2k}

Subtracting the above two inequalities, we have the following:

    f(x_i)-f(x)-\frac{1}{k}<q(x_i)-q(x)<f(x_i)-f(x)+\frac{1}{k}

The above inequality shows that for each i=1,\cdots,n, q(x_i) -q(x) \in U_i. Hence q-q(x) \in U. It is clear that q(x) \in V. This completes the proof that the map h is continuous.

The inverse is continuous

We now show that h^{-1} is continuous. Let (g,t) \in Y \times \mathbb{R}. Note that h^{-1}(g,t)=g+t. Let M be an open set in C_p(X) such that g+t \in M and

    M=\left\{f \in C_p(X): \forall \ i=1,\cdots,n+1, f(x_i) \in M_i \right\}

    \forall \ i=1,\cdots,n, \  M_i=(g(x_i)+t-\frac{1}{m},g(x_i)+t+\frac{1}{m})

    x_{n+1}=x

    M_{n+1}=(t-\frac{1}{m},t+\frac{1}{m})

where x_1,\cdots,x_n are arbitrary points of X and m is some large positive integer. Now define an open subset G \times T of Y \times \mathbb{R} such that (g,t) \in G \times T and

    G=\left\{q \in Y: \forall \ i=1,\cdots,n+1, q(x_i) \in G_i \right\}

    \forall \ i=1,\cdots,n, \  G_i=(g(x_i)-\frac{1}{2m},g(x_i)+\frac{1}{2m})

    T=(t-\frac{1}{2m},t+\frac{1}{2m})

We need to show that h^{-1}(G \times T) \subset M. Let (q,a) \in G \times T. We then have the following inequalities.

    \forall \ i=1,\cdots,n, \ g(x_i)-\frac{1}{2m}<q(x_i)<g(x_i)+\frac{1}{2m}

    t-\frac{1}{2m}<a<t+\frac{1}{2m}

Adding the above two inequalities, we obtain:

    \forall \ i=1,\cdots,n, \ g(x_i)+t-\frac{1}{m}<q(x_i)+a<g(x_i)+t+\frac{1}{m}

The above implies that \forall \ i=1,\cdots,n, q(x_i)+a \in M_i. It is clear that q(x_{n+1})+a=q(x)+a=a \in M_{n+1}. Thus q+a \in M. This completes the proof that h^{-1} is continuous.

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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 useful embedding for Cp(X)

Let X be a Tychonoff space (also called completely regular space). By C_p(X) we mean the space of all continuous real-valued functions defined on X endowed with the pointwise convergence topology. In this post we discuss a scenario in which a function space can be embedded into another function space. We prove the following theorem. An example follows the proof.

Theorem 1
Suppose that the space Y is a continuous image of the space X. Then C_p(Y) can be embedded into C_p(X).

Proof of Theorem 1
Let t:X \rightarrow Y be a continuous surjection, i.e., t is a continuous function from X onto Y. Define the map \psi: C_p(Y) \rightarrow C_p(X) by \psi(f)=f \circ t for all f \in C_p(Y). We show that \psi is a homeomorphism from C_p(Y) into C_p(X).

First we show \psi is a one-to-one map. Let f,g \in C_p(Y) with f \ne g. There exists some y \in Y such that f(y) \ne g(y). Choose some x \in X such that t(x)=y. Then f \circ t \ne g \circ t since (f \circ t)(x)=f(t(x))=f(y) and (g \circ t)(x)=g(t(x))=g(y).

Next we show that \psi is continuous. Let f \in C_p(Y). Let U be open in C_p(X) with \psi(f) \in U such that

    U=\left\{q \in C_p(X): \forall \ i=1,\cdots,n, \ q(x_i) \in U_i \right\}

where x_1,\cdots,x_n are arbitrary points of X and each U_i is an open interval of the real line \mathbb{R}. Note that for each i, f(t(x_i)) \in U_i. Now consider the open set V defined by:

    V=\left\{r \in C_p(Y): \forall \ i=1,\cdots,n, \ r(t(x_i)) \in U_i \right\}

Clearly f \in V. It follows that \psi(V) \subset U since for each r \in V, it is clear that \psi(r)=r \circ t \in U.

Now we show that \psi^{-1}: \psi(C_p(Y)) \rightarrow C_p(Y) is continuous. Let \psi(f)=f \circ t \in \psi(C_p(Y)) where f \in C_p(Y). Let G be open with \psi^{-1}(f \circ t)=f \in G such that

    G=\left\{r \in C_p(Y): \forall \ i=1,\cdots,m, \ r(y_i) \in G_i \right\}

where y_1,\cdots,y_m are arbitrary points of Y and each G_i is an open interval of \mathbb{R}. Choose x_1,\cdots,x_m \in X such that t(x_i)=y_i for each i. We have f(t(x_i)) \in G_i for each i. Define the open set H by:

    H=\left\{q \in \psi(C_p(Y)) \subset C_p(X): \forall \ i=1,\cdots,m, \ q(x_i) \in G_i \right\}

Clearly f \circ t \in H. Note that \psi^{-1}(H) \subset G. To see this, let r \circ t \in H where r \in C_p(Y). Now r(t(x_i))=r(y_i) \in G_i for each i. Thus \psi^{-1}(r \circ t)=r \in G. It follows that \psi^{-1} is continuous. The proof of the theorem is now complete. \blacksquare

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Example

The proof of Theorem 1 is not difficult. It is a matter of notating carefully the open sets in both function spaces. However, the embedding makes it easy in some cases to understand certain function spaces and in some cases to relate certain function spaces.

Let \omega_1 be the first uncountable ordinal, and let \omega_1+1 be the successor ordinal to \omega_1. Furthermore consider these ordinals as topological spaces endowed with the order topology. As an application of Theorem 1, we show that C_p(\omega_1+1) can be embedded as a subspace of C_p(\omega_1). Define a continuous surjection g:\omega_1 \rightarrow \omega_1+1 as follows:

    g(\gamma) = \begin{cases} \omega_1 & \mbox{if } \ \gamma =0 \\ \gamma-1 & \mbox{if } \ 1 \le \gamma < \omega \\ \gamma & \mbox{if } \ \omega \le \gamma < \omega_1  \end{cases}

The map g is continuous from \omega_1 onto \omega_1+1. By Theorem 1, C_p(\omega_1+1) can be embedded as a subspace of C_p(\omega_1). On the other hand, C_p(\omega_1) cannot be embedded in C_p(\omega_1+1). The function space C_p(\omega_1+1) is a Frechet-Urysohn space, which is a property that is carried over to any subspace. The function C_p(\omega_1) is not Frechet-Urysohn. Thus C_p(\omega_1) cannot be embedded in C_p(\omega_1+1). A further comparison of these two function spaces is found in this subsequent post.

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\copyright \ 2014 \text{ by Dan Ma}

Cp(X) is countably tight when X is compact

Let X be a completely regular space (also called Tychonoff space). If X is a compact space, what can we say about the function space C_p(X), the space of all continuous real-valued functions with the pointwise convergence topology? When X is an uncountable space, C_p(X) is not first countable at every point. This follows from the fact that C_p(X) is a dense subspace of the product space \mathbb{R}^X and that no dense subspace of \mathbb{R}^X can be first countable when X is uncountable. However, when X is compact, C_p(X) does have a convergence property, namely C_p(X) is countably tight.

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Tightness

Let X be a completely regular space. The tightness of X, denoted by t(X), is the least infinite cardinal \kappa such that for any A \subset X and for any x \in X with x \in \overline{A}, there exists B \subset A for which \lvert B \lvert \le \kappa and x \in \overline{B}. When t(X)=\omega, we say that Y has countable tightness or is countably tight. When t(X)>\omega, we say that X has uncountable tightness or is uncountably tight. Clearly any first countable space is countably tight. There are other convergence properties in between first countability and countable tightness, e.g., the Frechet-Urysohn property. The notion of countable tightness and tightness in general is discussed in further details here.

The fact that C_p(X) is countably tight for any compact X follows from the following theorem.

Theorem 1
Let X be a completely regular space. Then the function space C_p(X) is countably tight if and only if X^n is Lindelof for each n=1,2,3,\cdots.

Theorem 1 is the countable case of Theorem I.4.1 on page 33 of [1]. We prove one direction of Theorem 1, the direction that will give us the desired result for C_p(X) where X is compact.

Proof of Theorem 1
The direction \Longleftarrow
Suppose that X^n is Lindelof for each positive integer. Let f \in C_p(X) and f \in \overline{H} where H \subset C_p(X). For each positive integer n, we define an open cover \mathcal{U}_n of X^n.

Let n be a positive integer. Let t=(x_1,\cdots,x_n) \in X^n. Since f \in \overline{H}, there is an h_t \in H such that \lvert h_t(x_j)-f(x_j) \lvert <\frac{1}{n} for all j=1,\cdots,n. Because both h_t and f are continuous, for each j=1,\cdots,n, there is an open set W(x_j) \subset X with x_j \in W(x_j) such that \lvert h_t(y)-f(y) \lvert < \frac{1}{n} for all y \in W(x_j). Let the open set U_t be defined by U_t=W(x_1) \times W(x_2) \times \cdots \times W(x_n). Let \mathcal{U}_n=\left\{U_t: t=(x_1,\cdots,x_n) \in X^n \right\}.

For each n, choose \mathcal{V}_n \subset \mathcal{U}_n be countable such that \mathcal{V}_n is a cover of X^n. Let K_n=\left\{h_t: t \in X^n \text{ such that } U_t \in \mathcal{V}_n \right\}. Let K=\bigcup_{n=1}^\infty K_n. Note that K is countable and K \subset H.

We now show that f \in \overline{K}. Choose an arbitrary positive integer n. Choose arbitrary points y_1,y_2,\cdots,y_n \in X. Consider the open set U defined by

    U=\left\{g \in C_p(X): \forall \ j=1,\cdots,n, \lvert g(y_j)-f(y_j) \lvert <\frac{1}{n} \right\}.

We wish to show that U \cap K \ne \varnothing. Choose U_t \in \mathcal{V}_n such that (y_1,\cdots,y_n) \in U_t where t=(x_1,\cdots,x_n) \in X^n. Consider the function h_t that goes with t. It is clear from the way h_t is chosen that \lvert h_t(y_j)-f(x_j) \lvert<\frac{1}{n} for all j=1,\cdots,n. Thus h_t \in K_n \cap U, leading to the conclusion that f \in \overline{K}. The proof that C_p(X) is countably tight is completed.

The direction \Longrightarrow
See Theorem I.4.1 of [1].

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Remarks

As shown above, countably tightness is one convergence property of C_p(X) that is guaranteed when X is compact. In general, it is difficult for C_p(X) to have stronger convergence properties such as the Frechet-Urysohn property. It is well known C_p(\omega_1+1) is Frechet-Urysohn. According to Theorem II.1.2 in [1], for any compact space X, C_p(X) is a Frechet-Urysohn space if and only if the compact space X is a scattered 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 - 2015 \text{ by Dan Ma}

Cp(omega 1 + 1) is not normal

In this and subsequent posts, we consider C_p(X) where X is a compact space. Recall that C_p(X) is the space of all continuous real-valued functions defined on X and that it is endowed with the pointwise convergence topology. One of the compact spaces we consider is \omega_1+1, the first compact uncountable ordinal. There are many interesting results about the function space C_p(\omega_1+1). In this post we show that C_p(\omega_1+1) is not normal. An even more interesting fact about C_p(\omega_1+1) is that C_p(\omega_1+1) does not have any dense normal subspace [1].

Let \omega_1 be the first uncountable ordinal, and let \omega_1+1 be the successor ordinal to \omega_1. The set \omega_1 is the first uncountable ordinal. Furthermore consider these ordinals as topological spaces endowed with the order topology. As mentioned above, the space \omega_1+1 is the first compact uncountable ordinal. In proving that C_p(\omega_1+1) is not normal, a theorem that is due to D. P. Baturov is utilized [2]. This theorem is also proved in this previous post.

For the basic working of function spaces with the pointwise convergence topology, see the post called Working with the function space Cp(X).

The fact that C_p(\omega_1+1) is not normal is established by the following two points.

  • If C_p(\omega_1+1) is normal, then C_p(\omega_1+1) has countable extent, i.e. every closed and discrete subspace of C_p(\omega_1+1) is countable.
  • There exists an uncountable closed and discrete subspace of C_p(\omega_1 +1).

We discuss each of the bullet points separately.

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The function space C_p(\omega_1+1) is a dense subspace of \mathbb{R}^{\omega_1}, the product of \omega_1 many copies of \mathbb{R}. According to a result of D. P. Baturov [2], any dense normal subspace of the product of \omega_1 many separable metric spaces has countable extent (also see Theorem 1a in this previous post). Thus C_p(\omega_1+1) cannot be normal if the second bullet point above is established.

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Now we show that there exists an uncountable closed and discrete subspace of C_p(\omega_1 +1). For each \alpha with 0<\alpha<\omega_1, define h_\alpha:\omega_1 + 1 \rightarrow \left\{0,1 \right\} by:

    h_\alpha(\gamma) = \begin{cases} 1 & \mbox{if } \gamma \le \alpha \\ 0 & \mbox{if } \alpha<\gamma \le \omega_1  \end{cases}

Clearly, h_\alpha \in C_p(\omega_1 +1) for each \alpha. Let H=\left\{h_\alpha: 0<\alpha<\omega_1 \right\}. We show that H is a closed and discrete subspace of C_p(\omega_1 +1). The fact that H is closed in C_p(\omega_1 +1) is establish by the following claim.

Claim 1
Let h \in C_p(\omega_1 +1) \backslash H. There exists an open subset U of C_p(\omega_1 +1) such that h \in U and U \cap H=\varnothing.

First we get some easy cases out of the way. Suppose that there exists some \alpha<\omega_1 such that h(\alpha) \notin \left\{0,1 \right\}. Then let U=\left\{f \in C_p(\omega_1 +1): f(\alpha) \in \mathbb{R} \backslash \left\{0,1 \right\} \right\}. Clearly h \in U and U \cap H=\varnothing.

Another easy case: If h(\alpha)=0 for all \alpha \le \omega_1, then consider the open set U where U=\left\{f \in C_p(\omega_1 +1): f(0) \in \mathbb{R} \backslash \left\{1 \right\} \right\}. Clearly h \in U and U \cap H=\varnothing.

From now on we can assume that h(\omega_1+1) \subset \left\{0,1 \right\} and that h is not identically the zero function. Suppose Claim 1 is not true. Then h \in \overline{H}. Next observe the following:

    Observation.
    If h(\beta)=1 for some \beta \le \omega_1, then h(\alpha)=1 for all \alpha \le \beta.

To see this, if h(\alpha)=0, h(\beta)=1 and \alpha<\beta, then define the open set V by V=\left\{f \in C_p(\omega_1 +1): f(\alpha) \in (-0.1,0.1) \text{ and } f(\beta) \in (0.9,1.1) \right\}. Note that h \in V and V \cap H=\varnothing, contradicting that h \in \overline{H}. So the above observation is valid.

Now either h(\omega_1)=1 or h(\omega_1)=0. We claim that h(\omega_1)=1 is not possible. Suppose that h(\omega_1)=1. Let V=\left\{f \in C_p(\omega_1 +1): f(\omega_1) \in (0.9,1.1) \right\}. Then h \in V and V \cap H=\varnothing, contradicting that h \in \overline{H}. It must be the case that h(\omega_1)=0.

Because of the continuity of h at the point x=\omega_1, of all the \gamma<\omega_1 for which h(\gamma)=1, there is the largest one, say \beta. Now h(\beta)=1. According to the observation made above, h(\alpha)=1 for all \alpha \le \beta. This means that h=h_\beta. This is a contradiction since h \notin H. Thus Claim 1 must be true and the fact that H is closed is established.

Next we show that H is discrete in C_p(\omega_1 +1). Fix h_\alpha where 0<\alpha<\omega_1. Let W=\left\{f \in C_p(\omega_1 +1): f(\alpha) \in (0.9,1.1) \text{ and } f(\alpha+1) \in (-0.1,0.1) \right\}. It is clear that h_\alpha \in W. Furthermore, h_\gamma \notin W for all \alpha < \gamma and h_\gamma \notin W for all \gamma <\alpha. Thus W is open such that \left\{h_\alpha \right\}=W \cap H. This completes the proof that H is discrete.

We have established that H is an uncountable closed and discrete subspace of C_p(\omega_1 +1). This implies that C_p(\omega_1 +1) is not normal.

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Remarks

The set H=\left\{h_\alpha: 0<\alpha<\omega_1 \right\} as defined above is closed and discrete in C_p(\omega_1 +1). However, the set H is not discrete in a larger subspace of the product space. The set H is also a subset of the following \Sigma-product:

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

Because \Sigma(\omega_1) is the \Sigma-product of separable metric spaces, it is normal (see here). By Theorem 1a in this previous post, \Sigma(\omega_1) would have countable extent. Thus the set H cannot be closed and discrete in \Sigma(\omega_1). We can actually see this directly. Let \alpha<\omega_1 be a limit ordinal. Define t:\omega_1 + 1 \rightarrow \left\{0,1 \right\} by t(\beta)=1 for all \beta<\alpha and t(\beta)=0 for all \beta \ge \alpha. Clearly t \notin C_p(\omega_1 +1) and t \in \Sigma(\omega_1). Furthermore, t \in \overline{H} (the closure is taken in \Sigma(\omega_1)).

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Reference

  1. Arhangel’skii, A. V., Normality and Dense Subspaces, Proc. Amer. Math. Soc., 48, no. 2, 283-291, 2001.
  2. Baturov, D. P., Normality in dense subspaces of products, Topology Appl., 36, 111-116, 1990.

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\copyright \ 2014 \text{ by Dan Ma}