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$\textbf{Proposition}$ Let $(X,\mathcal{U})$ be a uniform space. Then the following are equivalent.

1. $(X,\mathcal{U})$ is uniformly normal.

2. If $C_{1},C_{2}\subseteq X$ are disjoint closed sets, then there is a uniformly continuous map $f:X\rightarrow[0,1]$ with $f|_{C_{1}}=0$ and $f|_{C_{2}}=1$.

3. $X$ is normal and every continuous map $f:X\rightarrow[0,1]$ is uniformly continuous.

4. $X$ is normal, and for every compact Hausdorff space $C$, every uniformly continuous map $f:X\rightarrow C$ is continuous.

$\textbf{Proposition}$ Let $(X,\mathcal{U})$ be a uniform space. Then the following are equivalent.

1. $(X,\mathcal{U})$ is uniformly normal.

2. If $C_{1},C_{2}\subseteq X$ are disjoint closed sets, then there is a uniformly continuous map $f:X\rightarrow[0,1]$ with $f|_{C_{1}}=0$ and $f|_{C_{2}}=1$.

3. $X$ is normal and every continuous map $f:X\rightarrow[0,1]$ is uniformly continuous.

4. $X$ is normal, and for every compact Hausdorff space $C$, every continuous map $f:X\rightarrow C$ is uniformly continuous.

$\textbf{Proposition}$ Let $(X,\mathcal{U})$ be a uniform space. Then the following are equivalent.

1. $(X,\mathcal{U})$ is uniformly normal.

2. If $C_{1},C_{2}\subseteq X$ are disjoint closed sets, then there is a uniformly continuous map $f:X\rightarrow[0,1]$ with $f|_{C_{1}}=0$ and $f|_{C_{2}}=1$.

3. $X$ is normal and every uniformly continuous map $f:X\rightarrow[0,1]$ is continuous.

4. $X$ is normal, and for every compact Hausdorff space $C$, every uniformly continuous map $f:X\rightarrow C$ is continuous.

$\textbf{Proposition}$ Let $(X,\mathcal{U})$ be a uniform space. Then the following are equivalent.

1. $(X,\mathcal{U})$ is uniformly normal.

2. If $C_{1},C_{2}\subseteq X$ are disjoint closed sets, then there is a uniformly continuous map $f:X\rightarrow[0,1]$ with $f|_{C_{1}}=0$ and $f|_{C_{2}}=1$.

3. $X$ is normal and every continuous map $f:X\rightarrow[0,1]$ is uniformly continuous.

4. $X$ is normal, and for every compact Hausdorff space $C$, every uniformly continuous map $f:X\rightarrow C$ is continuous.

$\textbf{Proposition}$ Let $X$ be a normal space. Then the following are equivalent.

1. Every compatible uniformity on $X$ is uniformly normal.

2. The space $X$ has a unique compatible proximity.

3. The space $X$ has a unique compactification.

4. $X$ is locally compact and the one-point-compactification of $X$ coincides with the Stone-Cech compactification of $X$.

5. $X$ is locally compact and $A\cap B\neq\emptyset$ whenever $A,B$ are non-compact closed subsets of $X$.

6. $X$ has a unique compatible uniformity.

$\textbf{Proposition}$ Let $X$ be a normal space. Then the following are equivalent.

1. Every compatible uniformity on $X$ is uniformly normal.

2. The space $X$ has a unique compatible proximity.

3. The space $X$ has a unique compactification.

4. $X$ is compact or $X$ is locally compact and the one-point-compactification of $X$ coincides with the Stone-Cech compactification of $X$.

5. $X$ is locally compact and $A\cap B\neq\emptyset$ whenever $A,B$ are non-compact closed subsets of $X$.

6. $X$ has a unique compatible uniformity.