# Is uniform continuity a property of the category of completely regular spaces?

If $(X, U)$ and $(Y, V)$ are uniform spaces then one has the notion of a map $f : X \to Y$ to be uniformly continuous relative to $U$ and $V$. A uniform space $(X, U)$ induces a completely regular topology $\tau_X$ on $X$ but $\tau_X$ can also be induced by other uniformities $U'$ on $X$.

Questions:

1. If $U$, $U'$ are uniformities on $X$ that both induce the same (completely regular) topology $\tau_X$ on $X$ and $V$ and $V'$ are uniformities on $Y$ that both induce the same (completely regular) topology $\tau_Y$ on $Y$ does it follow that a map $f : X \to Y$ that is uniformly continuous relative to $U$ and $V$ is also uniformly continuous relative to $U'$ and $V'$?

2. If 1. has a negative answer, can those uniform spaces $(Y,V)$ be characterized such that 1. has a positive answer for every completely regular space $(X, \tau_X)$? In other words, let $(Y, V)$ be a uniform space, $(X, \tau_X)$ a completely regular space and assume that whenever $U$ and $U'$ induce $\tau_X$ then a map $f : X \to Y$ is uniformly continuous relative to $U$ and $V$ if and only if $f$ is uniformly continuous relative to $U'$ and $V$. Can something be said about $(Y, V)$? Similarly, if we "fix" a uniform space $(X, U)$, can something be said about $(Y, \tau_Y)$ such that uniform continuity of $f : X \to Y$ is a property of the topology $\tau_Y$ and the uniformity $U$ on $X$?

3. If 1. has a negative answer, can those completely regular spaces $(X, \tau_X)$ be characterized such that uniform continuity of $f : X \to Y$ to any uniform space $(Y, V)$ depends only on the topology $\tau_X$, i.e. whenever $\tau_X$ is determined by uniformities $U$ and $U'$ on $X$ then $f$ is uniformly continuous relative to $U$ and $V$ if and only if $f$ is uniformly continuous relative to $U'$ and $V$? (And similarly as in 2., what happens if we switch the roles of the domain $X$ and the range $Y$, i.e. $X$ is uniform space and $Y$ is completely regular space.)

• No idea about 2 and 3, but as you seem to suspect the answer to 1 is negative. Metrics induce uniformities and there are many metrics on $\mathbb{R}$ with very different ideas of uniform continuity, a simple example being the metric $|x^3-y^3|$, which certainly makes the cube function uniformly continuous. But the topology induced by the cube metric is still just generated by open intervals. – Kevin Carlson Jan 25 '16 at 20:56
• There are some trivialised cases, e.g. if a space has a unique uniformity (that induces the topology). Such spaces are called "almost compact" by some (Gillman and Jerrison mention them). I believe they are characterised by the property that $\left|\beta(X)\setminus X\right|\le 1$; a non-compact example is $\omega_1$ in the order topology. – Henno Brandsma Jan 25 '16 at 21:26