This is a basic discussion on products of compact spaces. Let be a product space and let . The product of the spaces about the fixed point is the following subspace of the product space :
The products had been introduced in this blog. In a previous post, it was shown that the product of uncountably many separable spaces is an example of a space that has the countable chain condition (ccc) that is not separable. In this post, we discuss the product of compact spaces. First, the product of uncountably many spaces can never be paracompact because it always contains a closed copy of the first uncountable ordinal . Second, the product of compact spaces may not be normal. Our example is that is not normal where each . While it may be too much to expect that the Tychonoff theorem (the product space is compact if and only if each factor is compact) would have a counterpart for products, we do have a proof that the product of compact spaces is countably compact. This provides another example of a countably compact space that is non-compact. It is known that the product of metric spaces is normal ( and ).
When the index set is countable, the product is the entire product space . When all but countably many are one-point sets, the product can also be treated as just a product space of countably many factors. So we like to avoid these two situations by assuming that there are uncountably many spaces and each has at least two points. When this is the case, the product is a proper subspace of the associated product space. Such products are called proper.
For each , define by for each in the product space . The map maps each point in the product space to its coordinate and is called the projection map of on . Open sets of the form , where is finite and for each , is open in , form a base in the product topology.
Observation. Given and , it is easy to verify that is homeomorphic to . We will make use of this observation in the rest of this note.
Lemma. Let be a family of spaces each of which has at least two points. Let . Then (about the fixed point ) contains a closed copy of the first uncountable ordinal . Thus any proper product can never be paracompact.
Proof. For each , choose a point such that . Consider the point such that for each . Since each is Hausdorff, choose disjoint open sets containing and respectively.
We can think of elements of as all the functions such that for each and for all but countably many . For each , consider the function such that for all and for all . In other words, agrees with on the initial segment and agrees with on the final segment . Then is a closed copy of .
Claim 1. The set is closed in .
We show is open. Let . If for some , , choose open such that and does not contain both . Then and . Now we can assume that for each , we have . Since , there are such that and . Then and .
Claim 2. The mapping is a homeomorphism between and .
First show that the mapping is continuous. Let be an open set containing where is finite and is open for each . Let be an open interval in the order topology of such that misses . For each , it is clear that . Now, we show that the inverse is continuous. Consider the open interval . Note that . For each , we have .
Example. The space where each is not normal.
Let . Consider the product about the fixed point where each . Based on the observation above, is homeomorphic to . In turn is homeomorphic to . Based on the lemma, contains a closed copy of . It can be shown that is not normal (see a proof in this blog). Thus contains the non-normal closed subspace and is thus not normal.
Theorem 1. Let be a family of compact spaces. Then is countably compact.
Proof. Let be a countably infinite set. We show that has an accumulation point in . For each , let be the countable set on which the fixed point . Since is countable, is countable. We can consider as a subspace of . Since is compact, has an accumulation point in . Extend by letting on . It follows that is an accumulation point of .
- Gul’ko, S. P., On the properties of sets lying in products, Dokl. Acad. Nauk. SSSR, 237, (1977) 505-508 (in Russian).
- Rudin, M. E., products of metric spaces are normal, Preprint, 1977.