In his seminal 1937 paper, Jones [1] proved the following result about [Moore spaces](https://en.wikipedia.org/wiki/Moore_space_(topology)):



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> **Theorem. (Jones)** If $2^{\aleph\_0}<2^{\aleph\_1}$ then all *separable* normal Moore spaces are metrizable.
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Then he came up with the idea that maybe the *separability* condition in the above theorem could be removed. This led him to formulate the following famous conjecture:



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> **Normal Moore Space Conjecture (NMSC).** All normal Moore spaces are metrizable. 
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Long story short, NMSC turned out to be of high large cardinal strength and so far beyond the power of $ZFC$ to decide. 


In 1980, Nyikos [2] proved the consistency of NMSC using Product Measure Extension Axiom (PMEA) which is known to be consistent assuming the consistency of strongly compact cardinals. Later a more direct proof of the consistency of NMSC has been presented by Dow *et al* [3] using the stronger assumption of the consistency of supercompact cardinals. 


Interestingly, Fleissner [4] used a core model argument to prove the fact that the consistency of NMSC implies the consistency of measurable cardinals as well. Thus NMSC lies between strongly compacts and measurables in the *consistency strength* order of the large cardinals hierarchy. 


I wonder whether the *direct implication* power of NMSC is as strong as its large cardinal strength or not. 



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> **Question 1.** Does NMSC directly imply the existence of any large cardinals, say those of some topological nature such as weakly compacts? To be more precise, let me add that I look for the theorems of the form NMSC $\Rightarrow \exists \kappa$ Large (i.e. at least strongly inaccessible) if there is any. 
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**Update.** According to Will's answer, it turned out that NMSC doesn't contain any direct large cardinal strength. 


In other direction, one may strengthen NMSC as follows:



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> **Complete Normal Moore Space Conjecture (CNMSC).** All normal Moore spaces are *[completely metrizable](https://en.m.wikipedia.org/wiki/Completely_metrizable_space)*. 
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Here is my second question:



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> **Question 2.** Is CNMSC consistent? What are upper and lower bounds for its large cardinal strength? Strongly compacts and measurables or more?
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**References.**


1. *Jones, F. Burton*, [**Concerning normal and completely normal spaces**](http://dx.doi.org/10.1090/S0002-9904-1937-06622-5), Bull. Am. Math. Soc. 43, 671-677 (1937). [ZBL0017.42902](https://zbmath.org/?q=an:0017.42902).
2. *Nyikos, Peter J.*, [**A provisional solution to the normal Moore space problem**](http://dx.doi.org/10.2307/2042338), Proc. Am. Math. Soc. 78, 429-435 (1980). [ZBL0446.54030](https://zbmath.org/?q=an:0446.54030).
3. *Dow, Alan; Tall, Franklin D.; Weiss, William A. R.*, [**New proofs of the consistency of the normal Moore space conjecture. I & II**](http://dx.doi.org/10.1016/0166-8641(90)90013-R), Topology Appl. 37, No. 1, 33-51 (1990). [ZBL0719.54038](https://zbmath.org/?q=an:0719.54038).
4. *Fleissner, William G.*, [**If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal**](http://dx.doi.org/10.2307/1999211), Trans. Am. Math. Soc. 273, 365-373 (1982). [ZBL0498.54025](https://zbmath.org/?q=an:0498.54025).



On the Large Cardinal Strength of Normal Moore Space Conjecture
If $\text{NMSC}$ is consistent, then so is $\text{NMSC}+\text{"there are no strongly inaccessible cardinals"}$. 


This is because if $V \models \text{NMSC}$, then $V\_\kappa \models \text{NMSC}$ for any inaccessible cardinal $\kappa$. (A proof sketch is given below.) In particular, if $\kappa$ is the first strongly inaccessible cardinal, then $V\_\kappa \models \text{NMSC}+\text{"there are no strongly inaccessible cardinals"}$. 


[If you think through the [definition](https://en.wikipedia.org/wiki/Moore_space_(topology)) of a Moore space, it's not too hard to see that if $(X,\tau)$ is some topological space in $V\_\alpha$, then the statement "$(X,\tau)$ is a normal Moore space" can be decided by looking at $V\_{\alpha+\omega}$. Likewise, the fact that $(X,\tau)$ is metrizable can be decided by looking at $V\_{\alpha+\omega}$. Thus the truth/falsity of the NMSC for $(X,\tau)$ is the same in $V$ and in $V\_\kappa$. Because this is so for every topological space, the truth/falsity of the NMSC is the same in $V$ and in $V\_\kappa$.]


