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Definition: Let $X$ be a set. A set $U\subseteq X\times X$ is called entourage of the diagonal if $\Delta=\{(x,x):x\in X\}\subseteq U$ and $U=U^{-1}.$

Let $\Phi$ be a family of entourages of the diagonal. The pair $(X,\Phi)$ is called uniform space if for all entourages $U,V$ the following conditions are satisfied:

$1.U\in \Phi,U\subseteq V \implies V\in \Phi.$

$2.U,V\in \Phi\implies U\cap V\in \Phi.$

$3.U\in \Phi \implies (\exists V\in \Phi)V\circ V=\{(x,z):(\exists y\in X),(x,y),(y,z)\in V\}\subseteq U.$

$4.\bigcap\Phi=\Delta.$

The sets of the form $$U[x]=\{y\in X:(x,y)\in U\}$$ gives a local base at $x$ for the topology induced by the uniformity $\Phi.$

Now in this paper $I^K$-cauchy Functions by Sleziak,Toma,Das it says The last condition of the Definition implies that the induced topology is Hausdorff.

Now by last condition it means the $4$th one $\bigcap \Phi = \Delta.$

Let $x,y\in X;x\neq y.$ We need to find at least two $U\cap V=\phi,U,V\in \Phi$ such that $x\in U,y\in V.$Now according to the definitions wlg $$U=U[x]\\ V=V[y]$$. Now if they have a point say $(a,b)$ in common then to be in $U[x]$, $a=x$ and to be in $V[y]$ $a=y$ and thus making $x=y$ which contradicts our initial assumption that $x\neq y.$ So they are Hausdorff. But what has condition $4$ to do with it $?$

Also , from the proof , did we just find out that whenever two points are distinct in an Uniform Space all their nbd's are disjoint?

Thank You.

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    $\begingroup$ That is not the usual definition of uniform space: the usual definition does not require that $\bigcap\Phi=\Delta$ and does require that if $U\in\Phi$, then $U^{-1}\in\Phi$. The usual definition does not imply that a uniform space is Hausdorff. It does imply that a uniform space is completely regular; $\bigcap\Phi=\Delta$ immediately implies that it’s $T_1$, hence Tikhonov, hence Hausdorff. $\endgroup$ – Brian M. Scott Dec 18 '16 at 23:46
  • $\begingroup$ @BrianM.Scott: It's the definition I got from the paper I mentioned above. $\endgroup$ – user118494 Dec 19 '16 at 5:42
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    $\begingroup$ Yes, I understood that; I’m just warning you for future reference that it’s not the usual one. $\endgroup$ – Brian M. Scott Dec 19 '16 at 17:58
  • $\begingroup$ @BrianM.Scott I somehow missed the above comment when this was posted. We followed the definition from Engelking's book. In the paper, we explicitly mentioned that different definitions are also used. (That's what I have to say to our defense.) Anyway, +1 for your comment. $\endgroup$ – Martin Sleziak May 16 '17 at 8:40
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Let $x,y\in X$ and assume that every neighborhood of $x$ intersects every neighborhood of $y$. In particular we'll have: $$\forall U\in\Phi,\ \exists z_U\in X,\ z_U\in U[x]\cap U[y].$$ Which can be translated as $$\forall U\in\Phi,\ \exists z_U\in X,\ (x,z_U),(y,z_U)\in U.$$ Now, since $U$ is an entourage of $\Delta$, $U=U^{-1}$, hence we conclude: $$\forall U\in\Phi,\ \exists z_U\in X,\ (x,z_U),(z_U,y)\in U.$$

We now proceed by contradiction: assume that $x\neq y$, then, by Condition 4, there exists $U\in\Phi$ such that $(x,y)\not\in U$. By Condition 3, there exists $V\in\Phi$ such that $V\circ V\subset U$. But we have a contradiction with the fact that $(x,z_V),(z_V,y)\in V$ hence $(x,y)\in V\circ V\subset U$.


You went wrong there:

Now if they have a point say $(a,b)$ in common

This is nonsense: $U[x]$ is a subset of $X$, not of $X\times X$.


did we just find out that whenever two points are distinct in an Uniform Space all their nbd's are disjoint?

Certainly not!

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  • $\begingroup$ To say that "every neighborhood of $x$ intersects every neighborhood of $y$" does not mean what you say it means. Your quantifiers are wrong, the point $z$ will depend on the neighborhood. The correct statement is "for every $U,V \in \Phi$ there exists $z \in U[x] \cap V[y]$". $\endgroup$ – Lee Mosher Dec 18 '16 at 18:27
  • $\begingroup$ @LeeMosher: hopefully fixed… $\endgroup$ – gniourf_gniourf Dec 18 '16 at 18:45
  • $\begingroup$ So is our $U$ fixed here? $\endgroup$ – user118494 Dec 18 '16 at 19:48
  • $\begingroup$ @user118494: where? I usually leave no free variables, so things should be clear. $\endgroup$ – gniourf_gniourf Dec 18 '16 at 20:03
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If $x \neq y$, then $(x,y) \notin \Delta$, so if (4) holds, there is some $U \in \Phi$ such that $(x,y) \notin U$. Now let $V \in \Phi$ be such that $V \circ V \subseteq U$.

Now suppose $z \in V[x] \cap V[y]$. Then $(x,z) \in V, (z,y) \in V$ and symmetry yields that $(x,y) \in V \circ V$ so $(x,y) \in U$, which is a contradiction with how $U$ was chosen. So $V[x]$ and $V[y]$ are disjoint neighbourhoods for $x$ and $y$.

Note the abstract similarity with the proof for metrics: $d(x,y) = r > 0$ and we use a two balls of radius $\frac{r}{2}$ (Which corresponds to $V[x], V[y]$) and the triangle inequality then does the rest. Uniformities are inspired on metrics, so this often helps.

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