# Topology of uniform convergence of functions $X\to X$?

Suppose that $$X$$ is a compact Hausdorff space and that $$\phi\colon X\to X$$ is a homeomorphism. What does it mean to take the closure of $$\{\phi^{k}:k\in\mathbb{Z}\}$$ with respect to the topology of uniform convergence? Or more generally, what does the the topology of uniform convergence look like in this particular case? Since there is no metric involved, I don't know what 'uniform' means in this context. I read something about uniform spaces sitting properly between metric spaces and topological spaces, but it seemed quite technical.

$$X$$ is compact Hausdorff so it has a unique uniformity $$\mathcal{U}$$. It's convergence in the uniformity on $$C(X,X)$$ induced by $$\mathcal{U}$$ that is meant.

i.e. for all open neighbourhoods $$O$$ of $$\{(x,x): x \in X\}=:\Delta_X \subseteq X \times X$$ we have a basic open set $$B(f,O)$$ for $$f \in C(X,X)$$ defined by

$$B(f,O):= \{g \in C(X,X)\mid \forall x \in X: (f(x), g(x)) \in O\}$$

and this defines a local base at $$f$$ in the uniform topology. So now you know $$f$$ is in the closure of $$\{ \phi^k\mid k \in \Bbb Z\}$$ iff

$$\forall O \in \mathcal{N}(\Delta_X): \exists k \in \Bbb Z: \forall x \in X: (\phi^k(x), f(x)) \in O$$

This makes it quite concrete.

• +1. I was unaware of this simple fact. Commented May 13, 2021 at 11:42
• @KaviRamaMurthy not everyone knows about uniformities (though a general topologist can really profit from knowledge of it). They've gone out of style. Commented May 13, 2021 at 11:44
• @HennoBrandsma Thanks for your clear answer. Do you have any introductory references for uniform spaces? E.g. a reference that discusses why compact Hausdorff spaces admit a unique uniform structure. Commented May 13, 2021 at 14:24
• @Calculix Willard discusses this fact and uniform spaces in general too. Commented May 13, 2021 at 14:30