This is another post that discusses what is like when is a compact space. In this post, we discuss the example where is the first compact uncountable ordinal. Note that is the successor to , which is the first (or least) uncountable ordinal. The function space is monolithic and is a Frechet-Urysohn space. Interestingly, the first property is possessed by for all compact spaces . The second property is possessed by all compact scattered spaces. After we discuss , we discuss briefly the general results for .
The function space is a dense subspace of the product space . In fact, is homeomorphic to a subspace of the following subspace of :
The subspace is the -product of many copies of the real line . The -product of separable metric spaces is monolithic (see here). The -product of first countable spaces is Frechet-Urysohn (see here). Thus has both of these properties. Since the properties of monolithicity and being Frechet-Urysohn are carried over to subspaces, the function space has both of these properties. The key to the discussion is then to show that is homeopmophic to a subspace of the -product .
Connection to -product
We show that the function space is homeomorphic to a subspace of the -product of many copies of the real lines. Let be the following subspace of :
Every function in has non-zero values at only countably points of . Thus can be regarded as a subspace of the -product .
By Theorem 1 in this previous post, , i.e, the function space is homeomorphic to the product space . On the other hand, the product can also be regarded as a subspace of the -product . Basically adding one additional factor of the real line to still results in a subspace of the -product. Thus we have:
Thus possesses all the hereditary properties of . Another observation we can make is that is not hereditarily normal. The function space is not normal (see here). The -product is normal (see here). Thus is not hereditarily normal.
A closer look at
In fact has a stronger property that being monolithic. It is strongly monolithic. We use homeomorphic relation in (1) above to get some insight. Let be a homeomorphism from onto . For each , let be defined as follows:
Clearly . Furthermore can be considered as a subspace of and is thus metrizable. Let be a countable subset of . Then for some . The set is metrizable. The set is also a closed subset of . Then is contained in and is therefore metrizable. We have shown that the closure of every countable subspace of is metrizable. In other words, every separable subspace of is metrizable. This property follows from the fact that is strongly monolithic.
Monolithicity and Frechet-Urysohn property
As indicated at the beginning, the -product is monolithic (in fact strongly monolithic; see here) and is a Frechet-Urysohn space (see here). Thus the function space is both strongly monolithic and Frechet-Urysohn.
Let be an infinite cardinal. A space is -monolithic if for any with , we have . A space is monolithic if it is -monolithic for all infinite cardinal . It is straightforward to show that is monolithic if and only of for every subspace of , the density of equals to the network weight of , i.e., . A longer discussion of the definition of monolithicity is found here.
A space is strongly -monolithic if for any with , we have . A space is strongly monolithic if it is strongly -monolithic for all infinite cardinal . It is straightforward to show that is strongly monolithic if and only if for every subspace of , the density of equals to the weight of , i.e., .
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 . It is clear that all metrizable spaces are strongly monolithic.
The function space is not separable. Since it is strongly monolithic, every separable subspace of is metrizable. We can see this by knowing that is a subspace of the -product , or by using the homeomorphism as in the previous section.
For any compact space , is countably tight (see this previous post). In the case of the compact uncountable ordinal , has the stronger property of being Frechet-Urysohn. A space is said to be a Frechet-Urysohn space (also called a Frechet space) if for each and for each , if , then there exists a sequence such that the sequence converges to . As we shall see below, is rarely Frechet-Urysohn.
For any compact space , is monolithic but does not have to be strongly monolithic. The monolithicity of follows from the following theorem, which is Theorem II.6.8 in .
Then the function space is monolithic if and only if is a stable space.
See chapter 3 section 6 of  for a discussion of stable spaces. We give the definition here. A space is stable if for any continuous image of , the weak weight of , denoted by , coincides with the network weight of , denoted by . In , is notated by . The cardinal function is the minimum cardinality of all , the weight of , for which there exists a continuous bijection from onto .
All compact spaces are stable. Let be compact. For any continuous image of , is also compact and , since any continuous bijection from onto any space is a homeomorphism. Note that always holds. Thus implies that . Thus we have:
Let be a compact space. Then the function space is monolithic.
However, the strong monolithicity of does not hold in general for for compact . As indicated above, is monolithic but not strongly monolithic. The following theorem is Theorem II.7.9 in  and characterizes the strong monolithicity of .
Let be a space. Then is strongly monolithic if and only if is simple.
A space is -simple if whenever is a continuous image of , if the weight of , then the cardinality of . A space is simple if it is -simple for all infinite cardinal numbers . Interestingly, any separable metric space that is uncountable is not -simple. Thus is not -simple and is not strongly monolithic, according to Theorem 3.
For compact spaces , is rarely a Frechet-Urysohn space as evidenced by the following theorem, which is Theorem III.1.2 in .
Let be a compact space. Then the following conditions are equivalent.
- is a Frechet-Urysohn space.
- is a k-space.
- The compact space is a scattered space.
A space is a scattered space if for every non-empty subspace of , there exists an isolated point of (relative to the topology of ). Any space of ordinals is scattered since every non-empty subset has a least element. Thus is a scattered space. On the other hand, the unit interval with the Euclidean topology is not scattered. According to this theorem, cannot be a Frechet-Urysohn space.
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.