Let be a completely regular space (also called Tychonoff space). If is a compact space, what can we say about the function space , the space of all continuous real-valued functions with the pointwise convergence topology? When is an uncountable space, is not first countable at every point. This follows from the fact that is a dense subspace of the product space and that no dense subspace of can be first countable when is uncountable. However, when is compact, does have a convergence property, namely is countably tight.
Let be a completely regular 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. Clearly any first countable space is countably tight. There are other convergence properties in between first countability and countable tightness, e.g., the Frechet-Urysohn property. The notion of countable tightness and tightness in general is discussed in further details here.
The fact that is countably tight for any compact 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 the countable case of Theorem I.4.1 on page 33 of . We prove one direction of Theorem 1, the direction that will give us the desired result for where is compact.
Proof of Theorem 1
Suppose that is Lindelof for each positive integer. Let and where . For each positive integer , we define an open cover of .
Let be a positive integer. Let . Since , there is an such that for all . Because both and are continuous, for each , there is an open set with such that for all . Let the open set be defined by . Let .
For each , choose be countable such that is a cover of . Let . Let . Note that is countable and .
We now show that . Choose an arbitrary positive integer . Choose arbitrary points . Consider the open set defined by
We wish to show that . Choose such that where . Consider the function that goes with . It is clear from the way is chosen that for all . Thus , leading to the conclusion that . The proof that is countably tight is completed.
See Theorem I.4.1 of .
As shown above, countably tightness is one convergence property of that is guaranteed when is compact. In general, it is difficult for to have stronger convergence properties such as the Frechet-Urysohn property. It is well known is Frechet-Urysohn. According to Theorem II.1.2 in , for any compact space , is a Frechet-Urysohn space if and only if the compact space is a scattered space.
- Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.