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Alan Dow and Frank Tall recently proved the consistency of the statement Every hereditarily normal manifold of dimension at least two is metrizable.

See: Dow, Alan; Tall, Franklin D., Hereditarily normal manifolds of dimension greater than one may all be metrizable, Trans. Am. Math. Soc. 372, No. 10, 6805-6851 (2019). ZBL1427.54006.

This suggests a natural question:

Is it true in ZFC that every monotonically normal manifold of dimension at least two is metrizable?

The Long Line is an example of a one-dimensional non-metrizable monotonically normal manifold.

NOTE: By "manifold" I mean a Hausdorff space which is locally euclidean.

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    $\begingroup$ Dow and Tall's result doesn't use a supercompact. The supercompact was used in Nyikos' earlier result. $\endgroup$ Oct 30, 2020 at 18:17
  • $\begingroup$ Ok, I see, none of the consequences of PFA(S)[S] they're using requires large cardinals. $\endgroup$ Oct 30, 2020 at 20:07

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It would appear that that it is true. The result is due to Z. Balogh and E. Rudin and appears in their paper Monotone Normality, Top. App. 47, (1992), 115-127. The statement to quote is the following.

Corollary 2.3.(e). A manifold of dimension $\geq2$ is metrizable if and only if it is monotonically normal.

This follows easily from the first of the two main results in the paper.

Theorem I. A monotonically normal space is paracompact if and only if it does not have a closed subspace homeomorphic to a stationary subset of a regular uncountable cardinal.

It's worth mentioning that with a little thought it's not hard to repackage the dimension hypothesis so as to include all manifolds.

A manifold is metrisable if and only if it is separable and monotonically normal.

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