Category:Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods

This category contains pages concerning Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods:

Let $\struct{S, \tau}$ be a topological space.

For each $x \in S$, let:

$\map \UU x$ denote the system of open neighborhoods of $x$

Then $\struct{S, \tau}$ is a sober space if and only if:

for each completely prime filter $\FF$ in the complete lattice $\struct{\tau, \subseteq}$:

$\exists ! x \in S : \FF = \map \UU x$