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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$\copyright \ 2014 \text{ by Dan Ma}$

Basic topological properties of Corson compact spaces

A compact space is said to be a Corson compact space if it can be embedded in a $\Sigma$-product of real lines. Corson compact spaces play an important role in functional analysis. Corson compact spaces are also very interesting from a topological point of view. In this post, we discuss several basic topological properties of Corson compact spaces, some of which have been discussed and proved in some previous posts.

Links to other posts on Corson compact spaces are given throughout this post (as relevant results are discussed). A listing of these previous posts is also given at the end of this post.

For any infinite cardinal number $\kappa$, the $\Sigma$-product of $\kappa$ many copies of $\mathbb{R}$ is the following subspace of the product space $\mathbb{R}^\kappa$:

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

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

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Some examples of Corson compact spaces

When $\kappa=\omega$, $\Sigma(\omega)$ is simply $\mathbb{R}^\omega$, the product of countably many copies of the real lines. Any compact metrizable space can be embedded in $\mathbb{R}^\omega$; see Theorem 4.2.10 in [3]. Thus any compact metrizable space is Corson compact.

One easily described non-metrizable Corson compact space is the one-point compactification of an uncountable discrete space. Let $\tau$ be an uncountable cardinal number. Let $D_\tau$ be the discrete space of cardinality $\tau$. Let $p$ be a point not in $D_\tau$. Let $A_\tau=D_\tau \cup \left\{p \right\}$. Consider a topology on $A_\tau$ such that $D_\tau$ is discrete as before and any open neighborhood of $p$ has the form $\left\{p \right\} \cup C$ where $C \subset D_\tau$ and $D_\tau-C$ is finite. The space $A_\tau$ is better known as the one-point compactification of a discrete space of cardinality $\tau$. To see that $A_\tau$ is embedded in $\Sigma(\tau)$, for each $\alpha<\tau$, define $f_\alpha:\tau \rightarrow \left\{0,1 \right\}$ such that $f_\alpha(\alpha)=1$ and $f_\alpha(\beta)=0$ for all $\beta \ne \alpha$. Furthermore, let $g:\tau \rightarrow \left\{0,1 \right\}$ be defined by $g(\beta)=0$ for all $\beta<\tau$. It is easy to clear that $K=\left\{f_\alpha: \alpha<\tau \right\} \cup \left\{g \right\}$ is a subspace of $\Sigma(\tau)$. Note that $\left\{f_\alpha: \alpha<\tau \right\}$ is a discrete space of cardinality $\tau$ and that any open neighborhood of $g$ contains all but finitely many $f_\alpha$. Thus $K$ is homeomorphic to $A_\tau$.

Another non-metrizable Corson compact space is defined and discussed in this previous post.

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Some basic operations

Corson compact spaces behave nicely with respect to some natural topological operations such as the following:

1. Corson compactness is hereditary with respect to closed subspaces.
2. Corson compactness is preserved by taking continuous images.
3. Corson compactness is preserved by taking countable products.

It is clear that closed subspaces of every Corson compact space are also Corson compact. The fact that any continuous image of a Corson compact space is Corson compact (bullet point #2) is established in Theorem 6.2 of [6].

We prove bullet point #3, that the Tychonoff product of countably many Corson compact spaces is Corson compact. Let $K_1,K_2,K_3,\cdots$ be compact spaces such that each $K_j$ is a subspace of $\Sigma(\tau_j)$ for some infinite cardinal $\tau_j$. We show that the product $\prod_{j=1}^\infty K_j$ is Corson compact. Let $\tau$ be the cardinality of $\bigcup_{j=1}^\infty \tau_j$. Note that $\prod_{j=1}^\infty K_j$ is a compact subspace of $\prod_{j=1}^\infty \Sigma(\tau_j)$. In turn, $\prod_{j=1}^\infty \Sigma(\tau_j)$ is identical to $\Sigma(\tau)$. Thus $\prod_{j=1}^\infty K_j$ is Corson compact.

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Basic properties inherited from $\Sigma$-products

Some properties of Corson compact spaces are inherited from the $\Sigma$-products of real line. For example, we have the following two facts about $\Sigma$-products.

• The $\Sigma$-product of first countable spaces is a Frechet-Urysohn space.
• The $\Sigma$-product of separable metric spaces is monolithic.

Both of the above two properties are hereditary. Thus we have the following:

1. Every Corson compact space is a Frechet-Urysohn space.
2. Every Corson compact space is monolithic.

A space $X$ is monolithic if for each subspace $Y$ of $X$, the density of $Y$ coincides with the network weight of $Y$, i.e., $d(Y)=nw(Y)$. A space $X$ is strongly monolithic if for each subspace $Y$ of $X$, the density of $Y$ coincides with the weight of $Y$, i.e., $d(Y)=w(Y)$. Monolithic spaces are discussed in this previous post. For compact spaces, the notion of being monolithic and the notion of being strongly monolithic coincide. One obvious consequence of bullet point #5 is that being separable is an indicator of metrizability among Corson compact spaces. The following bullet point captures this observation.

1. Let $X$ be a Corson compact space. Then $X$ is metrizable if and only if $X$ is separable. See Proposition 1 in this previous post.

A space $Y$ is said to be Frechet space (also called Frechet-Urysohn 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 $y$. Thus any compact space that is not a Frechet-Urysohn space is not Corson compact. A handy example is the compact space $\omega_1+1$ with the order topology. Note that $\omega_1+1$ is monolithic. Thus monolithic compact spaces need not be Corson compact.

An extreme example of a compact non-Frechet-Urysohn space is one that has no non-trivial convergent sequence. For example, take $\beta \omega$, the Stone-Cech compactification of a countable discrete space, which has no non-trivial convergent sequences at any point. Thus $\beta \omega$ is not Corson compact.

Every Corson compact space has a $G_\delta$ point. It then follows that every Corson compact space has a dense set of $G_\delta$ points (see this previous post). In a compact space, there is a countable local base at every $G_\delta$ point. Thus we have the following bullet point.

1. Every Corson compact space has a dense first countable subspace.

However, it is not true that every Corson compact space has a dense metrizable subspace. See Theorem 9.14 in [7] for an example of a first countable Corson compact space with no dense metrizable subspace.

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Remarks

The results indicated here and proved in the previous posts represent a small sample of results on Corson compact spaces (and just focusing on the topological aspects). Many results on Corson compact spaces and Eberlein compact spaces are very deep results. The chapter c-16 in [5] is a good introduction. Some of the results proven in this and other posts in this blog are mentioned in [5] without proof. Interesting characterizations of Corson compact spaces are presented in [4].

In closing, we mention one more property. The authors in [1] showed that for any Corson compact space $X$, the function space $C_p(X)$ with the pointwise convergence topology is a Lindelof space. Thus we have the following bullet point.

1. The function space $C_p(X)$ is Lindelof for every Corson compact space $X$.

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Blog posts on Corson compact spaces

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Reference

1. Alster, K., Pol, R., On function spaces of compact subspaces of $\Sigma$-products of the real line, Fund. Math., 107, 35-46, 1980.
2. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
3. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
4. Gruenhage, G., Covering properties on $X^2 \backslash \Delta$, W-sets, and compact subsets of $\Sigma$-products, Topology Appl., 17, 287-304, 1984.
5. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
6. Michael, E., Rudin, M. E., A note on Eberlein compacts, Pacific J. Math., 128, 149-156 1987.
7. Todorcevic, S., Trees and Linearly Ordered Sets, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 235-293, 1984.

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$\copyright \ 2014 \text{ by Dan Ma}$

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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$\copyright \ 2014 \text{ by Dan Ma}$

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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$\copyright \ 2014 \text{ by Dan Ma}$