Full strength of Hausdorff axiom

Question

Hi i am reading Topology by Munkres and there in one section under Hausdorff Spaces it is written that

One reason for our lack of interest in the $T_1$ axiom is the fact that many of the interesting theorems of topology require not just that axiom, but the full strength of the Hausdorff axiom.

My doubt is that what does he mean by the full strength of the Hausdorff axiom?. For reference i am writing the $T_1$ axiom definition as they've given:The condition that finite point sets be closed has been given a name called $T_1$ axiom.I am also attaching the screenshot where i have highlighted the part(Page 99)What are we missing here that he has to say "full strength of the Hausdorff axiom"?

Answer 1

The technical meaning of "full strength" here is that a theorem requires the full strength of the assumption that a space is Hausdorff if it is false for every space that is not Hausdorff. So it does not suffice to assume anything that is "weaker" than the Hausdorff assumption.

A generalization of Theorem 17.10, which comes immediately after the bit you quote, is an example of such a theorem: a space $X$ is Hausdorff if and only if every net converges to at most one point in $X$. The "if and only if" means exactly that this statement fails in every non-Hausdorff space.

Here is another example: using the fact that compact Hausdorff spaces are normal, together with Urysohn's lemma, you can show that a compact space $X$ is Hausdorff if and only if continuous functions $f : X \to \mathbb{R}$ separate points (meaning that if $x \neq y$ then there exists a continuous function $f : X \to \mathbb{R}$ such that $f(x) \neq f(y)$). So this result requires the full strength of the Hausdorff assumption among compact spaces, and is false for every compact non-Hausdorff space.

I am actually not sure if Munkres intends the technical meaning of "full strength," though. A weaker version of what he means is that it does not suffice to assume a weaker separation axiom like $T_1$.

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We can also give a precise technical meaning to the usage of "strength" and "weaker" here, which is often not made explicit. It refers to the following:

• an assumption $A$ is stronger than an assumption $B$ (and $B$ is weaker than $A$) if $A$ implies $B$, meaning that in every case that $A$ is true, $B$ is also true; and
• $A$ is strictly stronger than $B$ (and $B$ is strictly weaker than $A$) if $A$ strictly implies $B$, meaning that in every case that $A$ is true, $B$ is also true, and there is also at least one case in which $A$ is false and $B$ is true.

So to say that Hausdorff is a stronger assumption than $T_1$ is to say that it implies $T_1$, and to say that it is strictly stronger means that there exists a $T_1$ space which is not Hausdorff. "Strength" refers to the fact that stronger assumptions let you prove more facts, because you are proving them about a more restricted set of cases.

For another example of this usage of "strength," some statements are commonly proven using the axiom of choice but "do not require the full strength of this assumption," which formally means that there exists a model of ZF set theory in which the axiom of choice is false but these statements are still true. An example of such a statement is "a product of compact Hausdorff spaces is compact Hausdorff," which turns out to only require (and I think be equivalent to?) the ultrafilter lemma, which is known to be "strictly weaker" than the axiom of choice (which is equivalent to Tychonoff's theorem, that a product of compact spaces is compact).