Let \(F\) be a closed, orientable surface of genus \(g\), endowed with a metric of constant curvature and denote by \({\mathcal G}_g\) the set of closed geodesics on \(F\). Closed geodesics are considered as images of periodic maps from \(\mathbb{R}\) into \(F\). When \(F=S^2\), then \({\mathcal G}_0\) inherits a topology by the natural bijection between the set of the great circles on \(S^2\) and \(\mathbb{R} P^2\). This topology can be described as the one having the set \({\mathcal U}_\varepsilon(c)= \{c'\in{\mathcal G}_0\mid \delta_{c,\varepsilon} (c')\geq\text{length} (c)\}\), \(\varepsilon>0\), as fundamental system of neighborhoods of any \(c\in{\mathcal G}_0\). Here, \(\delta_{c,\varepsilon} (c')\) is defined as the length of an arc obtained as intersection of \(c'\in{\mathcal G}_0\) with the \(\varepsilon\)-neighborhood of \(c\) on \(S^2\). If \(F\) has genus \(g\geq 1\), the author describes a topology \({\mathcal T}\) on \({\mathcal G}_g\), slightly modifying the previous description passing through the universal covering of \(F\). He proves that the so obtained topology is Hausdorff when \(g=1\), \(F\) being the torus \(T^2\) with a flat metric, whereas it is not Hausdorff when \(g>1\), \(F\) being a surface with a fixed hyperbolic metric. Moreover, in the last case, one can consider the set \({\mathcal G}^0_g\) of all geodesics on \(F\), dropping the closing condition, and topologize it as follows: The preimage in \(\mathbb{H}^2\) of a geodesic on \(F\) is a set of geodesics which is invariant with respect to the covering transformation group \(\Gamma\). Then \({\mathcal G}^0_g\) is identified with the quotient space \(((S^1_\infty\times S^1_\infty -\Delta)/ \mathbb{Z}_2)/\Gamma\), where \(S^1_\infty\) is the sphere at infinity of \(\mathbb{H}^2\) and \((S^1_\infty\times S^1_\infty-\Delta)/ \mathbb{Z}_2\) is homeomorphic to an open Möbius strip. The author proves that the relative topology induced from \({\mathcal G}^0_g\) on \({\mathcal G}_g\) is equivalent to \({\mathcal T}\) and this implies that the space \({\mathcal G}^0_g= ((S^1_\infty\times S^1_\infty-\Delta)/ \mathbb{Z}_2)/ \Gamma\) is not a Hausdorff space for \(g>1\).