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This question is related to this one on MathOverflow, although somewhat different. It is also clearly related to a previous question of mine, although hopefully not a duplicate.

Question: The "usual" definition of manifold is (1) locally Euclidean, (2) Hausdorff, (3) second countable. Another commonly used, but weaker, definition is (1) locally Euclidean, (2) metrizable.

Is the following definition equivalent to the "usual" definition?

$$\text{(1) locally Euclidean (2) metrizable (3) separable}\iff \\ \text{(1) locally Euclidean (2) Hausdorff (3) second countable} ?$$

Attempt:

$\implies$ Obviously metrizable implies Hausdorff. Moreover, for metrizable spaces, separability and second countability are equivalent, so metrizable+separable implies second countable.

$\impliedby$ Locally Euclidean implies locally metrizable (since Euclidean space is metrizable). Locally Euclidean implies locally compact, and as in this question on Math.SE, locally compact+Hausdorff+second countable implies paracompact. Then a theorem of Smirnov says that locally metrizable+Hausdorff+paracompact implies metrizable.

Second countable always implies separable.

Note: This result doesn't really seem useful to me. It popped in my head the other day and I figured it would make sense to double-check and/or to let anyone else see it who might be interested.

Also, the above is also equivalent to:

$$\text{(1) locally Euclidean (2) metrizable (3) Lindelöf} $$

since, for metrizable spaces, separable $\iff$ second countable $\iff$ Lindelöf.

Also worth mentioning: the above is also equivalent to:

$$\text{(1) locally Euclidean (2) embeddable in some Euclidean space} $$

This follows in one direction from the fact that every subspace of a metrizable space is metrizable, and every subspace of a second countable space is second countable. The other direction is nontrivial, the Whitney embedding theorem.

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