Recently, I read the topology of Munkres. In the chapter $ 4 $, he presents some proofs of the metrization theorem of Urysohn then, at the end of the section, generalizes one of them to an “ incorporation theorem '' # 39; for completely regular spaces (Munkres definition of completely regular includes the $ T_0 $ axiom):

**Theorem 34.2 (nesting theorem).** Let $ X $ to be a space in which the sets of a point are closed. Assume that $ {f _ { alpha} } _ { alpha in J} $ is an indexed family of continuous functions $ f_a: X rightarrow mathbb R $ fulfilling the requirement that for each point $ x_0 $ of $ X $ and each neighborhood $ U $ of $ x_0 $, there is an index $ alpha $ such as $ f _ { alpha} $ is positive to $ x_0 $ and disappears outside $ U $. Then the function $ F: X rightarrow mathbb R ^ J $ Defined by $$ F (x) = left (f _ { alpha} (x) right) _ { alpha in J} $$

is a recess of $ X $ in $ mathbb R ^ J $. Yes $ f _ { alpha} $ maps $ X $ in $ (0.1) $ for each $ alpha $, then $ F $ impregnated $ X $ in $ (0,1) ^ J $.

My question is this: is the proof of this theorem (the way Munkres describes it above) implicitly based on the axiom of choice or on a weaker version? If the answer is "a weaker version", what is this version? Is the theorem itself equivalent to a well-known weaker version of $ AC $?

**preparation:**

The essential elements of the evidence – that $ F $ is injective, continuous and an open map, follow easily by simply plugging in $ J $ instead of the index set in the first proof of Urysohn's metrization theorem presented by Munkres.

The thing that has bothered a recent rereading of this evidence is the construction of the & # 39;$ F $& # 39; used in the statement of the nesting theorem itself.

It seems that, implicitly, in the construction of the function, & # 39;$ F $& # 39; we used the truth of a statement that would look something like:

Yes $ Y $ is a set and $ {Z _ { beta} } _ { beta in J} $ is an indexed family of s.t. sets for each $ beta in J $, $ exists $ some $ g _ { beta}: Y rightarrow Z _ { beta} $, then there is a function: $$ G: Y rightarrow prod _ { beta in J} Z _ { beta} $$ Defined by $$ G = left (g _ { beta} right) _ { beta in J} $$

But this statement looks, at least to me, a version of the Axiom of Choice.

For, with the truth of the Axiom of Choice, given these $ Y $, $ {Z _ { beta} } _ { beta in J} $, and $ {g _ { beta} } _ { beta in J} $, we can order the index set well, $ J $, then define recursively $ pi _ { beta} (G) = g _ { beta} $ (or $ pi _ { beta} $ denote it $ beta $projection map).

On the other hand, if the above statement was true, given any indexed family of non-empty sets, $ {X _ { alpha} } _ { alpha in J} $, we have this, for each $ alpha $, $ exists $ $ f _ { alpha}: {0 } rightarrow X _ { alpha} $ (since each $ X _ { alpha} $ is not empty – I think this collection of functions should be well defined). So by the truth of the statement, there is a map, $ F: {0 } rightarrow prod _ { alpha in J} X _ { alpha} $. But then $ prod _ { alpha in J} X _ { alpha} $ must contain $ F (0) $doing $ prod _ { alpha in J} X _ { alpha} $ not empty.

So the statement seems to imply the Axiom of Choice too, making it *look* as if they were equivalent. I must say, however, that I am not sure that the above implications are true. My set theory is rusty and I have not found and I do not remember ever seeing the Axiom of Choice spelled out this way, so I think there may be a fault in my reasoning.

So my question is – well, first, if there is a flaw in my reasoning establishing an equivalence between the statement and $ AC $, please leave a response explaining this – but beyond that, may the statement be equivalent to something weaker? If not, is there a way around $ AC $ in the proof of Imbedding's theorem? I ask mainly because I don't really know $ AC $ appearing in topology as versions of Tychonoff's theorem, the connection of which seems more intuitive.