Maybe someone familiar with Willard's textbook can help me out. Problem section 23G on pg. 174 is titled piecewise metrizability. The first problem is:
- If a Tychonoff space $X$ is the union of a locally finite collection of closed, metrizable subspaces, then X is metrizable.
This fact is well-known. It is 4.4.19 in Engelking's book. It also appears in Nagata's seminal metrization paper from 1950. However, the conclusion is true without the assumption that $X$ is Tychonoff. This leads me to think there is a simpler proof under this assumption. If you know of one, please give me an outline or hint.
More interesting is the next problem from Willard:
- If a $T_4$ space $X$ is the union of any locally finite collection of metrizable subspaces, then $X$ is metrizable. [Use 15.10]
This is less well-known, because it is clearly not true. Consider the one point compactification of an uncountable discrete space. It is normal and non-metrizable, but is the union of just two metrizable subspaces: one discrete and the other a singleton. The hint refers to the standard "shrinking" theorem for point-countable open covers of normal spaces. So, it seems likely that the word "open" was just accidentally omitted. The following fact follows easily from the first exercise and 15.10:
Fact: If a $T_4$ space $X$ is the union of a locally finite collection of open, metrizable subspaces, then $X$ is metrizable.
Certainly, someone must have noticed this before now. However, I haven't been able to find any reference to, or use of, the above fact in print. Does anyone know of a better citation for this than "Corrected version of exercise 2 in 23G of [W]"?
Thanks in advance,
- Willard, Stephen General topology Originally printed 1970 by AddisonWesley; currently in print by Dover Publications (2004)
- Engelking, Ryszard General topology Heldermann Verlag (1989)
- Nagata, Jun-iti On a necessary and sufficient condition of metrizability J. Inst. Polytech. Osaka City Univ. (1950)
PS: Unfortunately, exercises 1 and 2 also appear (without correction) in Patty's more recent textbook, Foundations of Topology, Jones and Bartlett Publishers (2009). They appear to have been copied verbatim from Willard.