Let be the first uncountable ordinal, and let be the successor ordinal to . Furthermore consider these ordinals as topological spaces endowed with the order topology. It is a well known fact that any continuous real-valued function defined on either or is eventually constant, i.e., there exists some such that the function is constant on the ordinals beyond . Now consider the function spaces and . Thus individually, elements of these two function spaces appear identical. Any matches a function where is the result of adding the point to where is the eventual constant real value of . This fact may give the impression that the function spaces and are identical topologically. The goal in this post is to demonstrate that this is not the case. We compare the two function spaces with respect to some convergence properties (countably tightness and Frechet-Urysohn property) as well as normality.
One topological property that is different between and is that of tightness. The function space is countably tight, while is not countably tight.
Let be a space. The tightness of , denoted by , is the least infinite cardinal such that for any and for any with , there exists for which and . When , we say that has countable tightness or is countably tight. When , we say that has uncountable tightness or is uncountably tight.
First, we show that the tightness of is greater than . For each , define such that for all and for all . Let be the function that is identically zero. Then where is defined by . It is clear that for any countable , . Thus cannot be countably tight.
The space is a compact space. The fact that is countably tight follows from the following theorem.
Let be a completely regular space. Then the function space is countably tight if and only if is Lindelof for each .
Theorem 1 is a special case of Theorem I.4.1 on page 33 of  (the countable case). One direction of Theorem 1 is proved in this previous post, the direction that will give us the desired result for .
The Frechet-Urysohn property
In fact, has a property that is stronger than countable tightness. The function space is a Frechet-Urysohn space (see this previous post). Of course, not being countably tight means that it is not a Frechet-Urysohn space.
The function space is not normal. If is normal, then would have countable extent. However, there exists an uncountable closed and discrete subset of (see this previous post). On the other hand, is Lindelof. The fact that is Lindelof is highly non-trivial and follows from . The author in  showed that if is a space consisting of ordinals such that is first countable and countably compact, then is Lindelof.
Embedding one function space into the other
The two function space and are very different topologically. However, one of them can be embedded into the other one. The space is the continuous image of . Let be a continuous surjection. Define a map by letting . It is shown in this previous post that is a homeomorphism. Thus is homeomorphic to the image in . The map is also defined in this previous post.
The homeomposhism tells us that the function space , though Lindelof, is not hereditarily normal.
On the other hand, the function space cannot be embedded in . Note that is countably tight, which is a hereditary property.
There is a mapping that is alluded to at the beginning of the post. Each is associated with which is obtained by appending the point to where is the eventual constant real value of . It may be tempting to think of the mapping as a candidate for a homeomorphism between the two function spaces. The discussion in this post shows that this particular map is not a homeomorphism. In fact, no other one-to-one map from one of these function spaces onto the other function space can be a homeomorphism.
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
- Buzyakova, R. Z., In search of Lindelof ‘s, Comment. Math. Univ. Carolinae, 45 (1), 145-151, 2004.