Uniform Spaces - Uniformly continuous

Question

How to show that any function from a Discrete uniform Space to any other uniform space is uniformly continuous?

Answer 1

A function $f: (X, \mathcal{U}_d) \to (Y, \mathcal{U})$ (supposing you use entourages) is uniformly continuous iff for every $U \in \mathcal{U}$, $(f \times f)^{-1}[U]\in \mathcal{U}_d$, which is clear as $\Delta_X \subseteq (f \times f)^{-1}[U]$ as $\Delta_Y \subseteq U$. So $f$ is uniformly continuous.

Answer 2

If $(X,\Phi_X)$ is a discrete uniform space, then$$\Phi_X=\left\{U\subset X\times X\,\middle|\,\Delta\subset U\right\},$$where $\Delta=\bigl\{(x,x)\,|\,x\in X\bigr\}$. Now, let $(Y,\Phi_Y)$ be any uniformspace and let $f\colon X\longrightarrow Y$ be any function. Then, for every $V\in\Phi_Y$, there is a $U\in\Phi_X$ such that$$\bigl(\forall(x_1,x_2)\in U\bigr):\bigl(f(x_1),f(x_2)\bigr)\in V,$$namely taking $U=\Delta$. Therefore, $f$ is uniformly continuous.