Compare Theorem 42.1 (the Smirnov metrization theorem) with Exercises 7 and 8 of §34.
Solution
The Smirnov metrization theorem says \[ X\text{ metrizable} \quad\Longleftrightarrow\quad X\text{ paracompact Hausdorff and locally metrizable}. \] Both criteria of §34 follow from it, because each hypothesis implies paracompactness.
In E-MQTBP, \(X\) is compact Hausdorff and locally metrizable. Every compact Hausdorff space is paracompact, so the Smirnov theorem gives metrizability.
In E-KK77N, \(X\) is regular, Lindelöf, and locally metrizable. Every regular Lindelöf space is paracompact: the construction in E-KVFCT(a) gives a locally finite open refinement of every open cover. So the Smirnov theorem gives metrizability.
Conversely, compactness with the Hausdorff property and regularity with the Lindelöf property are two sufficient conditions for the paracompactness hypothesis of the Smirnov theorem.