3.I.4A

Metric and Topological Spaces | Part IB, 2005

Show that a topology τ1\tau_{1} is determined on the real line R\mathbf{R} by specifying that a nonempty subset is open if and only if it is a union of half-open intervals {ax<b}\{a \leq x<b\}, where a<ba<b are real numbers. Determine whether (R,τ1)\left(\mathbf{R}, \tau_{1}\right) is Hausdorff.

Let τ2\tau_{2} denote the cofinite topology on R\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 map(R,τ1)(R,τ2)\operatorname{map}\left(\mathbf{R}, \tau_{1}\right) \rightarrow\left(\mathbf{R}, \tau_{2}\right).

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