Show that a topology $\tau_{1}$ is determined on the real line $\mathbf{R}$ by specifying that a nonempty subset is open if and only if it is a union of half-open intervals $\{a \leq x<b\}$, where $a<b$ are real numbers. Determine whether $\left(\mathbf{R}, \tau_{1}\right)$ is Hausdorff.

Let $\tau_{2}$ denote the cofinite topology on $\mathbf{R}$ (that is, a non-empty subset is open if and only if its complement is finite). Prove that the identity map induces a continuous $\operatorname{map}\left(\mathbf{R}, \tau_{1}\right) \rightarrow\left(\mathbf{R}, \tau_{2}\right)$.