When Is A Product of Metrizable Spaces Normal?

Let \mathbb{N} be the discrete space the natural numbers. It is known that \mathbb{N}^{\omega_1} is not normal (see a proof here). It turns out that the product of uncountably many non-compact metric spaces is never normal. In 1948, A. H. Stone proved that the uncountable product of metric spaces is normal if and only if all but countably many factors are compact. Thus product of uncountably many copies of \mathbb{R} is never normal.

Theorem. Let \lbrace{Y_\alpha:\alpha \in S}\rbrace be a family of metrizable spaces. The following conditions are equivalent.
(1) \Pi_{\alpha \in S}Y_\alpha is paracompact.
(2) \Pi_{\alpha \in S}Y_\alpha is normal.
(3) All but countably many Y_\alpha are compact.

Proof. (1) \rightarrow (2) is obvious.

(2) \rightarrow (3). Suppose Y=\Pi_{\alpha \in S}Y_\alpha is normal. I Claim that all but countably many factors are countably compact. When this claim is established, (3) is established. Note that if a paracompact space is countably compact, it is compact. Suppose that there are uncountably many Y_\alpha that are not countably compact where \alpha < \omega_1. Then each such Y_\alpha would contain a closed copy of \mathbb{N}. Thus the product space Y would contains \mathbb{N}^{\omega_1} as a closed subspace. This is a contradiction since any closed subspace of a normal space is normal. Thus all but countably many factors are countably compact.

(3) \rightarrow (1). Suppose all but countably many Y_\alpha are compact. Then Y=\Pi_{\alpha \in S}Y_\alpha=H \times G where H is the product of all the compact factors Y_\alpha and G is the product of the countably many non-compact factors. Note that G is also a metrizable space. The product of a compact space and a paracompact space is paracompact (see a proof here). Thus (1) is established.

Corollary. Let \lbrace{Y_\alpha:\alpha \in S}\rbrace be a family of separable metrizable spaces. The following conditions are equivalent.
(1) \Pi_{\alpha \in S}Y_\alpha is paracompact.
(2) \Pi_{\alpha \in S}Y_\alpha is Lindelof.
(3) \Pi_{\alpha \in S}Y_\alpha is normal.
(4) All but countably many Y_\alpha are compact.

The only thing I want to mention about the corollary is that being a product of separable spaces, the product space \Pi_{\alpha \in S}Y_\alpha has the countable chain condition (ccc). In any space with the ccc, paracompactness implies the Lindelof property (see a proof here).

Reference
[Stone] Stone, A. H., [1948] Paracompact and Product Spaces, Bull. Amer. Math. Soc., 54, 977-982.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s