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}

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