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.

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}

Every Corson compact space has a dense first countable subspace

In any topological space X, a point x \in X is a G_\delta point if the one-point set \left\{ x \right\} is the intersection of countably many open subsets of X. It is well known that any compact Hausdorff space is first countable at every G_\delta point, i.e., if a point of a compact space is a G_\delta point, then there is a countable local base at that point. It is also well known that uncountable power of first countable spaces can fail to be first countable at every point. For example, no point of the compact space [0,1]^{\omega_1} can be a G_\delta point. In this post, we show that any Corson compact space has a dense set of G_\delta point. Therefore, any Corson compact space is first countable on a dense set (see Corollary 4 below). However, it is not true that every Corson compact space has a dense metrizable subspace. See Theorem 9.14 in [2] for an example of a first countable Corson compact space with no dense metrizable subspace. A list of other blog posts on Corson compact spaces is given at the end of this post.

The fact that every Corson compact space has a dense first countable subspace is taken as a given in the literature. For one example, see chapter c-16 of [1]. Even though Corollary 4 is a basic fact of Corson compact spaces, the proof involves much more than a direct application of the relevant definitions. The proof given here is intended to be an online resource for any one interested in knowing more about Corson compact spaces.

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

For each x \in \Sigma(\kappa), let S(x) denote the support of the point x, i.e., S(x) is the set of all \alpha<\kappa such that x_\alpha \ne 0.

Proposition 1
Let Y be a Corson compact space. Then Y has a G_\delta point.

Proof of Proposition 1
If Y is finite, then every point is isolated and is thus a G_\delta point. Assume Y is infinite. Let \kappa be an infinite cardinal number such that Y \subset \Sigma(\kappa). For f,g \in Y, define f \le g if the following holds:

    \forall \ \alpha \in S(f), f(\alpha)=g(\alpha)

It is relatively straightforward to verify that the following three properties are satisfied:

  • f \le f for all f \in Y. (reflexivity)
  • For all f,g \in Y, if f \le g and g \le f, then f=g. (antisymmetry)
  • For all f,g,h \in Y, if f \le g and g \le h, then f \le h. (transitivity)

Thus \le as defined here is a partial order on the compact space Y. Let C \subset Y such that C is a chain with respect to \le, i.e., for all f,g \in C, f \le g or g \le f. We show that C has an upper bound (in Y) with respect to the partial order \le. We need this for an argument using Zorn’s lemma.

Let W=\bigcup_{f \in C} S(f). For each \alpha \in W, choose some f \in C such that \alpha \in S(f) and define u_\alpha=f_\alpha. For all \alpha \notin W, define u_\alpha=0. Because C is a chain, the point u is well-defined. It is also clear that f \le u for all f \in C. If u \in Y, then u is a desired upper bound of C. So assume u \notin Y. It follows that u is a limit point of C, i.e., every open set containing u contains a point of C different from u. Hence u is a limit point of Y too. Since Y is compact, u \in Y, a contradiction. Thus it must be that u \in Y. Thus every chain in the partially ordered set (Y,\le) has an upper bound. By Zorn’s lemma, there exists at least one maximal element with respect to the partial order \le, i.e., there exists t \in Y such that f \le t for all f \in Y.

We now show that t is a G_\delta point in Y. Let S(t)=\left\{\alpha_1,\alpha_2,\alpha_3,\cdots \right\}. For each p \in \mathbb{R} and for each positive integer n, let B_{p,n} be the open interval B_{p,n}=(p-\frac{1}{n},p+\frac{1}{n}). For each positive integer n, define the open set O_n as follows:

    O_n=(B_{t_{\alpha_1},n} \times \cdots \times B_{t_{\alpha_n},n} \times \prod_{\alpha<\kappa,\alpha \notin \left\{ \alpha_1,\cdots,\alpha_n \right\}} \mathbb{R}) \cap Y

Note that t \in \bigcap_{n=1}^\infty O_n. Because t is a maximal element, note that if g \in Y such that g_\alpha=t_\alpha for all \alpha \in S(t), then it must be the case that g=t. Thus if g \in \bigcap_{n=1}^\infty O_n, then g_\alpha=t_\alpha for all \alpha \in S(t). We have \left\{t \right\}= \bigcap_{n=1}^\infty O_n. \blacksquare

Lemma 2
Let Y be a compact space such that for every non-empty compact subspace K of Y, there exists a G_\delta point in K. Then every non-empty open subset of Y contains a G_\delta point.

Proof of Lemma 2
Let U_1 be a non-empty open subset of the compact space Y. If there exists y \in U_1 such that \left\{y \right\} is open in Y, then y is a G_\delta point. So assume that every point of U_1 is a non-isolated point of Y. By regularity, choose an open subset U_2 of Y such that \overline{U_2} \subset U_1. Continue in the same manner and obtain a decreasing sequence U_1,U_2,U_3,\cdots of open subsets of Y such that \overline{U_{n+1}} \subset U_n for each positive integer n. Then K=\bigcap_{n=1}^\infty \overline{U_n} is a non-empty closed subset of Y and thus compact. By assumption, K has a G_\delta point, say p \in K.

Then \left\{p \right\}=\bigcap_{n=1}^\infty W_n where each W_n is open in K. For each n, let V_n be open in Y such that W_n=V_n \cap K. For each n, let V_n^*=V_n \cap U_n, which is open in Y. Then \left\{p \right\}=\bigcap_{n=1}^\infty V_n^*. This means that p is a G_\delta point in the compact space Y. Note that p \in U_1, the open set we start with. This completes the proof that every non-empty open subset of Y contains a G_\delta point. \blacksquare

Proposition 3
Let Y be a Corson compact space. Then Y has a dense set of G_\delta points.

Proof of Proposition 3
Note that Corson compactness is hereditary with respect to closed sets. Thus every compact subspace of Y is also Corson compact. By Proposition 1, every compact subspace of Y has a G_\delta point. By Lemma 2, Y has a dense set of G_\delta points. \blacksquare

Corollary 4
Every Corson compact space has a dense first countable subspace.

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Blog posts on Corson compact spaces

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Reference

  1. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
  2. 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}

An example of a non-metrizable Corson compact space

A compact space is said to be a Corson compact space if it can be embedded in a \Sigma-product of real lines. All compact metric spaces are Corson. In this post, we present a non-metrizable example of a Corson compact space found in the literature, found in [1]. A listing of other blog posts on Corson compact spaces is 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 \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. 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 [2]. 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.

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Describing the example

We define a Corson compact space C that is not metrizable. To define the example, let W be a set of real numbers such that the cardinality of W is \omega_1. Let \ll be a well ordering on the set W such that for each y \in W, the initial segment S_y=\left\{x \in W: x \ll y \right\} is countable. In other words, the well-ordered set (W,\ll) is of type \omega_1. Let < be the usual order on the real numbers. Consider the following collection of subsets of W:

    \mathcal{H}=\left\{H \subset W: \text{ the order } < \text{ coincides with the well order } \ll \text{ on the set } H \right\}

Note that H \in \mathcal{H} if and only if this condition holds: if x,y \in W with x \ne y, then

    x<y if and only if x \ll y.

Thus by default, \varnothing \in \mathcal{H} and \left\{ x \right\} \in \mathcal{H} for any x \in W. More importantly, every H \in \mathcal{H} is a countable set. To see this, suppose that J \subset W is uncountable. Then J has a two-sided limit point, say p (see this previous post). It does not matter whether p \in J. What matters is that for any open interval (a,b) with p \in (a,b), both (a,p) and (p,b) contain points of J. Then there is a sequence \left\{x_n: n=1,2,3,\cdots \right\} of points of J converging to p and there is a sequence \left\{y_n: n=1,2,3,\cdots \right\} of points of J converging to p such that

    y_1<y_2< \cdots < y_n < y_{n+1} < \cdots p \cdots< x_{n+1} <x_n < \cdots < x_2<x_1

Suppose that (J,<) is a well-ordered set. Let t \in J such that t is the least number that is an upper bound of Y=\left\{y_n: n=1,2,3,\cdots \right\}. It must be the case that p \le t. If p=t, then p is an element of J that has no immediate successor, a contradiction. If p<t, then t is not the least upper bound of Y. It follows that (J,<) cannot be a well-ordered set. Then the orders < and \ll do not agree on the set J. Thus no uncountable set can be in the family \mathcal{H}.

Consider the compact product space X=\left\{0,1 \right\}^{W}. For each f \in X, let S(f) be the support of the point f, i.e., the set of all x \in W such that f(x) \ne 0. Consider the following subspace of X:

    C=\left\{f \in X: S(f) \in \mathcal{H} \right\}

Note that z \in C where z(x)=0 for all x \in W and that for each t \in W, f_t \in C where f_t(t)=1 and f_t(x)=0 for all x \in W-\left\{t \right\}. Since each H \in \mathcal{H} is countable, C is a subspace of the \Sigma-product \Sigma(\omega_1). Since C is a subspace of the compact space X=\left\{0,1 \right\}^{W}, in order to show that C is compact, we only need to show that C is closed in X. To this end, let g \in X-C. Then (S(g),<) and (S(g),\ll) do not coincide. Then there exist c,d \in S(g) such that c<d and c \ll d cannot both be true. Suppose c<d and c \not \ll d. Consider the following open subset of X:

    U=\left\{f \in X: f(c)=f(d)=1 \right\}

It is clear that U \cap C=\varnothing. Thus C is a closed subspace of X=\left\{0,1 \right\}^{W}. Since C is a compact space and is a subspace of a \Sigma-product of real lines, C is a Corson compact space.

If a compact space is metrizable then it is separable. Thus if we can show that C is not separable, then C is not metrizable. We show that no countable subspace of C can be dense in C. Let D be a countable subspace of C. Let S=\bigcup_{f \in D} S(f), which is countable. Choose x \in W-S. Then the open set \left\{g \in C: g(x)=1 \right\} contains no point of D. This completes the proof that C is a non-metrizable Corson compact space.

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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. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.

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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}