# When is a topological space uniquely uniformizable?

In general, multiple uniformities can induce a given topology. But if a topological space is a compact Hausdorff space, then there is only one uniformity which induces its topology. The formal way to say it is, compact Hausdorff spaces are uniquely uniformizable.

My question is, what other spaces are uniquely uniformizable? Are there specific examples of such spaces, and is there a general chatacteization of such spaces?

Engelking (in General topology, (2nd ed.)) mentions these spaces in exercises 8.5.11 and 8.5.12 with a reference to

Doss, R. On uniform spaces with a unique structure, Amer. J. Math. 71 (1949), 19-23.

Looking for that reference online I quickly found related papers and looking in online databases of papers could get you many more perhaps.

Ex. 8.5.11 states that such spaces $$X$$ are Tychonov spaces that have a unique Hausdorff compactification (up to equivalence).

Ex. 8.5.12. asks to show that in such spaces $$X$$, if they are not compact, can have no complete uniformity on $$X$$, with a reference to

Dieudonné, J. Sur les espaces uniformes complets, Ann. Sci. École Normale Sup. 56 (1939), 277-291.

Willard (General Topology) mentions them in exercise 41F where one has to prove that $$|\beta X - X| \le 1$$ is equivalent to having a unique uniformity (or proximity) on a Tychonoff space $$X$$. (Where $$\beta X$$ is the Čech-Stone compactification of $$X$$, as usual.)

This means that $$\omega_1$$, the first uncountable ordinal in its order topology, is an example of a compact and uniquely uniformisable space, as $$\beta \omega_1 = \omega_1 +1$$, and similarly for the long line, which has its one-point compactification as its Čech-Stone compactification. Or of course spaces like $$\beta X \setminus \{p\}$$ for some remainder point $$p \in \beta X$$.

Such spaces are also mentioned in the classic book by Gilman and Jerrison, Rings of continuous functions, how call such spaces almost compact in exercises 6J and 15R. The former states the equivalence in terms of the Čech-Stone compactification again, as Willard does while the latter has some more formulations for such $$X$$:

TFAE for a Tychonoff space $$X$$:

• $$X$$ has a unique compatible uniform structure.
• Every continuous mapping from $$X$$ into any uniform space $$Y$$ is uniformly continuous in every compatible uniformity on $$X$$.
• Every function in $$C(X)$$ is uniformly continuous in every compatible uniformity on $$X$$.
• Every function in $$C^\ast(X)$$ is uniformly continuous in every compatible uniformity on $$X$$.
• Every function in $$C^\ast(X)$$ is uniformly continuous in every precompact (definition in the book) compatible uniformity on $$X$$.
• $$X$$ admits only one precompact uniformity.
• $$X$$ has a unique compactification.