Geodesic
A Metrization Theorem for 2-Manifolds
Canadian mathematical bulletin, Tome 19 (1976) no. 1, pp. 95-104

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There are few known metrization theorems for manifolds (locally Euclidean, connected, Hausdorff space). It is well known that for manifolds metrizability, second countability, Lindelöf's condition, σ-compactness and paracompactness are equivalent. Although these conditions imply separability, the latter does not imply any of the former (see Example 2.2), as is often believed. A common source of metrization for a covering manifold is that lifted from the base manifold [8; p. 181].For 2-manifolds, the presence of a complex analytic structure gives us a metrization theorem; it has been shown [1] that such manifolds are topologically characterized as those which are orientable and second countable. 
Vincent, Paul A. A Metrization Theorem for 2-Manifolds. Canadian mathematical bulletin, Tome 19 (1976) no. 1, pp. 95-104. doi: 10.4153/CMB-1976-014-x
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     title = {A {Metrization} {Theorem} for {2-Manifolds}},
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