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Include the empty set, and take all unions of intervals $[a,b)$ with $a<b$. These intervals cover the real line, and the intersection of two is either empty or $[\max (a,c),\min (b,d))$. Distributing intersections over unions therefore makes finite intersections of open sets open; arbitrary unions are open by construction. The whole line is open, so the axioms of a topology hold. This is the lower limit topology, whose space is the Sorgenfrey line.

If $x<y$, the neighborhoods $[x,y)$ and $[y,y+1)$ are disjoint. Hence $\boxed{(\mathbb{R},{\tau }_1)\text{ is Hausdorff}}$.

For the identity into the cofinite topology, an inverse image is the same subset of $\mathbb{R}$. Let $F$ be finite and $x\notin F$. If $F$ has a point greater than $x$, choose $b$ to be its least such point; otherwise take $b=x+1$. Then $[x,b)$ avoids every point of $F$, including $b$ itself. Taking these intervals over all $x\notin F$ writes $\mathbb{R}\setminus F$ as a ${\tau }_1$-open set. The inverse image of every cofinite open set, and of the empty set, is open. The identity map is continuous, because ${\tau }_2\subseteq {\tau }_1$.