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It's worth noting that any $T_1$ space which admits partitions of unity for finite (two element even) covers is Hausdorff:

Proof: Let $x, y \in X$. Let $U = X \ {x}, V = X \ {y}$. Then let ${f, g}$ form a partition of unity with $f$ subordinate to $U$ and $g$ subordinate to $V$. Then $A = { t : f(t) > \frac{1}{2} }$ and $B = { t : g(t) > \frac{1}{2} }$ A and B are disjoint open sets with $y \in A$ and $x \in B$.

Edit: On closer inspection, this if of course just the standard proofs that the existence of partitions of unity for finite covers implies normality + the fact that $T_1$ normal spaces are hausdorff

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It's worth noting that any $T_1$ space which admits partitions of unity for finite (two element even) covers is Hausdorff:

Proof: Let $x, y \in X$. Let $U = X \ {x}, V = X \ {y}$. Then let ${f, g}$ form a partition of unity with $f$ subordinate to $U$ and $g$ subordinate to $V$. Then $A = { t : f(t) > \frac{1}{2} }$ and $B = { t : g(t) > \frac{1}{2} }$ A and B are disjoint open sets with $y \in A$ and $x \in B$.