The minimal Cauchy filters associated to points define a uniformly continuous dense canonical map

Define $\eta(x)$ to be the minimal filter associated to the principal filter $\mathcal P_x$ at $x$. Since $\eta(x)=m(\mathcal P_x)\subseteq\mathcal P_x$, every member of $\eta(x)$ contains $x$.

Let $\widehat E[\mathcal F]$ be a basic neighbourhood. Choose a symmetric $D$ with $D^{\circ2}\subseteq E$, a $D$-small $A\in\mathcal F$, and $a\in A$. The point filter $\eta(a)$ contains $D[a]$, and $D[a]\times A\subseteq D^{\circ2}\subseteq E$, so $\eta(a)\in\widehat E[\mathcal F]$. Thus every basic neighbourhood meets $\eta[X]$.

Given a target basic entourage $\widehat E$, choose a symmetric $D$ with $D^{\circ3}\subseteq E$. If $(x,y)\in D$, then $D[x]\in\eta(x)$ and $D[y]\in\eta(y)$, while $D[x]\times D[y]\subseteq D^{\circ3}\subseteq E$. Hence $(\eta(x),\eta(y))\in\widehat E$, which proves uniform continuity.

Thus every neighbourhood meets $\eta[X]$, so its closure is all of $\widehat X$ and the image is dense.