When can we choose a covering local basis at a point which does not contain the whole space? I was wondering if we can ensure by relatively mild seperation axioms that a local basis ,which is a cover of the space, for at a topology at a point $x_0\in X$ does not contain $X$?
I'm pretty sure that if $X$ is $T_1$ and has more, then $3$ points for any $y\in X$ distinct from $x_0$ we have an open neighbourhood of $x_0$ containing also $y$ which is not the whole space, by the following argument: 
I need to find for all $y\in X$ an open neighbourhood $U_y$ such that $x_0,y\in U_y$ and $U_y\neq X$. Let $z\in X$ be a point distinct from both $x_0$ and $y$. Then by the Hausdorff property, there exists $U_{y,z}$ which contains $y$ but not $z$. Similarly we have a $U_{x_0,z}$ containing $x_0$ but not $z$. Then $U_y:=U_{y,z}\cup U_{x_0,z}$ is such a neighbourhood.
My question is first whether the argument seems valid, and second can one reduce the assumption on the seperation axiom to obtain the same argument? I would also like to know whether it seems that $T_0$ is sufficient for having a basis, not local, which does not contain $X$?
Edit:
I added an assumption on that local basis would also be a basis, since that's what I had in mind and it seems that my original question was simpler and not what I had intended to ask about.  I would appreciate if any one could adress my assumptions now.

A point p in a space S can have a local base without S
iff there is an open nhood of p that is not S.  
The Serpenski space ({0,1}, {empty set, {0}, {0,1}})
is a T$_0$ space where the only local base for 1 is {{0,1}}.  
Exercise.  Show if open U is nhood p, then
{ V : p in V, V open subset U } is a local base for p. 
