Separation Axioms: is it true that $T_4 \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1 \Rightarrow T_0$?

Question

I am thoroughly confused by the separation axioms in topology. My lectures, online resources, and books all seem to say different things.

Online I read that $T_4 \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1 \Rightarrow T_0$, the proofs make sense to me and I don't see how this could be wrong. I am reading the book Topology: An Introduction, by Stefan Waldmann. Given the chain of implications from before as fact, several things he says in the book confuse me:

Let $(M, \mathcal{M})$ be a topological space.

• he says that $(M, \mathcal{M})$ is regular if it is $T_1$ and $T_3$. But $T_3 \Rightarrow T_2 \Rightarrow T_1$, so why have the condition that it is $T_1$ and what is the difference between a regular space and a $T_3$ space?
• $(M, \mathcal{M})$ is normal if it is $T_1$ and $T_4$. Again, why have $T_1$ in the definition and what is the difference between normal and $T_4$?
• He says a metric space is normal and hence $T_1, T_2, T_3, T_4$. Why did he not include $T_0$ in this list?

Further, the Wikipedia article "Separation Axiom" adds a whole new level of confusion. There it is implied that normal and regular spaces are not necessarily $T_0$.

So is $T_4 \Rightarrow T_3 \Rightarrow T_2 \Rightarrow T_1 \Rightarrow T_0$ even true? If not, why not?

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Here are the definitions of the axioms as I learned them in my lecture and the book:

Let $(M, \mathcal{M})$ be a topological space.

• The space $M$ is called a $T_0$-space if for each two different points $p \neq q$ in $M$ we find an open subset which contains only one of them.
• The space $M$ is called a $T_1$-space if for each two different points $p \neq q$ in $M$ we find open subsets $O_1$ and $O_2$ with $p \in O_1$ and $q \in O_2$ but $p \not\in O_2$ and $q \not\in O_1$.
• The space $M$ is called a $T_2$-space or a Hausdorff space if for each two different points $p \neq q$ in $M$ we find disjoint open subsets $O_1$ and $O_2$ with $p \in O_1$ and $q \in O_2$.
• The space $M$ is called a $T_3$-space if for every closed subset $A \subset M$ and every $p \in M \backslash A$ there are disjoint open subsets $O_1$ and $O_2$ with $A \subset O_1$ and $p \in O_2$.
• The space $M$ is called a $T_4$-space if for two disjoint closed subsets $A_1, A_2 \subset M$ there are disjoint open subsets $O_1$ and $O_2$ with $A_1 \subset O_1$ and $A_2 \subset O_2$.

Answer 1

From Munkres’ Topology, section 31:

A two-point space in the indiscrete topology satisfies the other parts of the definitions of regularity and normality, even though it is not Hausdorff.

There is a minor deviation in terminologies: Some authors adopt the notion of $_3$ (or regular) being “a point and a closed set (not including that point) can be separated”, while others write $_3$ to say the bold quoted sentence plus Hausdorff. Similar for $T_4$ and its synonym normal.

In terms of your lecture, a two-point space in the indiscrete topology is $T_3$ and $T_4$, but not $T_2$. Thus $T_3\Rightarrow T_2$ is not valid.

P.S. We have $T_3+T_1\Rightarrow T_2$ and $T_4+T_1\Rightarrow T_2$, so this example does not satisfy $T_1$ as well.

Answer 2

I personally prefer that the $T_i$ axioms form a decreasing hierarchy as you suggest. $T_2 \implies T_1 \implies T_0$ already holds and it makes sense to extend this into the "higher" ones.

This leaves the "word"-versions (normal, regular, completely regular, corresponding to $T_{3\frac12}$ inbetween $T_4$ and $T_3$) for the "pure" point and closed set and closed sets versions of the axioms.

But having normal but no (or hardly any) closed sets to separate (like for the indiscrete space) makes it a rather "empty" notion I think. Normal is useful as it's precisely what needed for Urysohn's lemma, so it's handy to have a separate term for that. Adding $T_1$ (or $T_0$) to normal and regular at least leaves us with some closed sets (singletons etc) to separate. It makes the property more "interesting" and achieves the desired implications I mentioned.

There are similar debates among topology texts whether to include Hausdorffness in "compact" or "paracompact" etc. (see "quasicompact").

It's good to be aware that people don't agree on terms and always look in the introductions of papers (or indexes of books) to see how a symbol $T_3$ or word like "regular" is used. There are different traditions and if you're not careful you can misquote (or wrongly use) a paper's result.

Make a personal choice and stick to it would be my advice.