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**](http://dx.doi.org/10.1090/tran/7916), Trans. Am. Math. Soc. 372, No. 10, 6805-6851 (2019). [ZBL1427.54006](https://zbmath.org/?q=an:1427.54006).


This suggests a natural question:



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> Is it true in ZFC that every [monotonically normal](https://en.wikipedia.org/wiki/Monotonically_normal_space) manifold of dimension at least two is metrizable?
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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.



Are all monotonically normal manifolds of dimension at least two metrizable?
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.



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> **Corollary 2.3.(e).** A manifold of dimension $\geq2$ is metrizable if and only if it is monotonically normal.
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This follows easily from the first of the two main results in the paper.



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



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