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[parent] separated uniform space (Definition)

Let $ X$ be a uniform space with uniformity $ \mathcal{U}$. $ X$ is said to be separated or Hausdorff if it satisfies the following separation axiom:

$\displaystyle \bigcap \mathcal{U}=\Delta,$
where $ \Delta$ is the diagonal relation on $ X$ and $ \bigcap \mathcal{U}$ is the intersection of all elements (entourages) in $ \mathcal{U}$. Since $ \Delta\subseteq \bigcap \mathcal{U}$, the separation axiom says that the only elements that belong to every entourage of $ \mathcal{U}$ are precisely the diagonal elements $ (x,x)$. Equivalently, if $ x\ne y$, then there is an entourage $ U$ such that $ (x,y)\notin U$.

The reason for calling $ X$ separated has to do with the following assertion:

$ X$ is separated iff $ X$ is a Hausdorff space under the topology $ T_{\mathcal{U}}$ induced by $ \mathcal{U}$.

Recall that $ T_{\mathcal{U}}=\lbrace A\subseteq X\mid$   for each $ x\in A$, there is $ U\in \mathcal{U}$, such that $ U[x]\subseteq A\rbrace$, where $ U[x]$ is some uniform neighborhood of $ x$ where, under $ T_{\mathcal{U}}$, $ U[x]$ is also a neighborhood of $ x$. To say that $ X$ is Hausdorff under $ T_{\mathcal{U}}$ is the same as saying every pair of distinct points in $ X$ have disjoint uniform neighborhoods.

Proof. $ (\Rightarrow)$. Suppose $ X$ is separated and $ x,y\in X$ are distinct. Then $ (x,y)\notin U$ for some $ U\in \mathcal{U}$. Pick $ V\in \mathcal{U}$ with $ V\circ V\subseteq U$. Set $ W=V\cap V^{-1}$, then $ W$ is symmetric and $ W\subseteq V$. Furthermore, $ W\circ W\subseteq V\circ V\subseteq U$. If $ z\in W[x]\cap W[y]$, then $ (x,z),(y,z)\in W$. Since $ W$ is symmetric, $ (z,y)\in W$, so $ (x,y)=(x,z)\circ (z,y)\in W\circ W\subseteq U$, which is a contradiction.

$ (\Leftarrow)$. Suppose $ X$ is Hausdorff under $ T_{\mathcal{U}}$ and $ (x,y)\in U$ for every $ U\in \mathcal{U}$ for some $ x,y\in X$. If $ x\ne y$, then there are $ V[x]\cap W[y]=\varnothing$ for some $ V,W\in \mathcal{U}$. Since $ (x,y)\in V$ by assumption, $ y\in V[x]$. But $ y\in W[y]$, contradicting the disjointness of $ V[x]$ and $ W[y]$. Therefore $ x=y$. $ \qedsymbol$



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Other names:  separating, Hausdorff uniform space

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Cross-references: contradiction, symmetric, disjoint, points, neighborhood, uniform neighborhood, topology, Hausdorff space, iff, diagonal, entourages, intersection, diagonal relation, Hausdorff, uniformity, uniform space
There are 13 references to this entry.

This is version 2 of separated uniform space, born on 2007-02-18, modified 2008-06-02.
Object id is 8925, canonical name is SeparatedUniformSpace.
Accessed 806 times total.

Classification:
AMS MSC: 54E15 (General topology :: Spaces with richer structures :: Uniform structures and generalizations)
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