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I can understand (somewhat) why one would want to study non-Hausdorff topologies, since for example the Zariski topology is so important to algebraists, and the weak topology generated by lower semicontinuous functions on the real line also isn't T2.

But I have never heard of any non-T1 (not Kolmogorov, points may not be closed) space which is useful.

Moreover, for any non-T1 space isn't there a quotient map to a T1 space, just using the equivalence relation "two points are equivalent if they are topologically equivalent, ie share the same neighborhood system"? (This might even be a homeomorphism?)

Why isn't T1 the fourth topological axiom? In fact, I believe I saw one author use it as such. In any case I don't see any motivation to consider spaces which don't satisfy the axiom. Am I correct in this?

EDIT: Uh-oh. I thought that Kolmogorov/T0 was the same thing as T1 (i.e. I forgot that there exists something in between Hausdorff and "all points topologically indistinguishable").

Should I vote to close this question since it was based on a false premise?

I am interested in the reasons for studying T0 and not T1 spaces, so on one hand I do want to leave it open.

On the other hand, what I was really confused about is why we shouldn't reduce all non-T0 spaces to T0 spaces using the above mentioned equivalence relation, and thus why anyone would study non-T0 spaces?

I thought that the condition that all points are topologically distinguishable was only strictly weaker than Hausdorff, but equivalent to "all points are closed" -- I now realize my mistake.

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    $\begingroup$ The quotient of a T1 space may not be T1. Also, with the Sierpiński space (that is, $\{0,1\}$ with the topology $\{\emptyset,\{1\},\{0,1\}\}$), the two points are not topologically equivalent, so quotienting out topological equivalence does nothing. $\endgroup$ – Akiva Weinberger Jun 10 '16 at 3:46
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    $\begingroup$ I asked about non-T^0 here: math.stackexchange.com/questions/986804/… $\endgroup$ – leewz Jan 25 '17 at 22:25
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since for example the Zariski topology is so important to algebraists

The Zariski topology is not in general $T_1$.

The Zariski topology on $\text{Spec}(R)$, the prime spectrum of a commutative ring $R$ is always $T_0$ but generally not $T_1$. The non-closed points correspond to prime ideals which are not maximal. They are important to the understanding of schemes.

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    $\begingroup$ @Mathemagician1234 When did they ask for an example other than the Zariski topology? $\endgroup$ – user223391 Jun 10 '16 at 4:12
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    $\begingroup$ @Mathemagician1234 The OP's only mention of the Zariski topology is as an example of an interesting non-$T_2$ space. Zachary's answer shows that the Zariski topology can be non-$T_1$ as well - so this provides a good answer to the question (the OP already understands that the Zariski topology is important, and the answer gives an example of when the Zariski topology can be non-$T_1$; admittedly there's a jump in abstraction, since the Zariski topology of an algebraic variety over an ACF is always $T_1$, but still - spectra of commutative rings aren't much of a jump, and are plenty useful). $\endgroup$ – Noah Schweber Jun 10 '16 at 4:25
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    $\begingroup$ @Mathemagician1234 That's not at all what Noah said. $\endgroup$ – user223391 Jun 10 '16 at 6:40
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You can find an interesting contribution to your question in K. H. Hofmann's article The Low Separation Axioms T0 and T1: the topics of interest cited in this survey are algebraic geometry, operator theory, directed completeness (Scott topology and the like) and injective $T_0$-spaces (used to find models of $\lambda$-claculus)

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There are several important examples of a $T_0$ space,which is Kolomogrov but not $T_1$. $T_0$ is the weakest separation condition that can be imposed on a topological space.

The Wikipedia page for Kolomogrov spaces gives several very good examples of $T_0$ but not $T_1$ such as:

1) The "particular point" topology on any set with at least two elements is T0 but not T1 since the particular point is not closed (its closure is the whole space). An important special case is the Sierpiński space which is the particular point topology on the set {0,1}.
2) The "excluded point" topology on any set with at least two elements is T0 but not T1. The only closed point is the excluded point.
3) The Alexandrov topology on a poset is T0 but will not be T1 unless the order is discrete (agrees with equality). Every finite T0 space is of this type. This also includes the particular point and excluded point topologies as special cases.
4) The "right order" topology on a totally ordered set is a related example.
5) The "overlapping interval" topology is similar to the particular point topology since every open set includes 0.

A $T_0$ space allows one to distinguish between points topologically without imposing any stronger conditions for topological properties, which allows us to study topological relations among singleton sets. The discrete topology allows singletons to be open, but it doesn't really allow topological distinctions between them because their complements are also open.

For example, in the particular point topology, closed sets have empty interiors. We can prove this as follows: Given an open set $A \subset\mathbb(X)$ every $x \ne p$ is a limit point of A. So the closure of any open set other than $\emptyset$ is $X$. No closed set other than $\mathbb(X)$ contains p so the interior of every closed set other than $\mathbb(X)$ is $\emptyset$.Also, in the particular point topology, the closure of a compact subspace is not itself compact. We prove it as follows: The set {p} is a compact space. However its closure is the entire space $\mathbb(X)$ and if $\mathbb(X)$ is infinite this is not compact (since any set {t,p} is open).

For similar reasons if $\mathbb(X)$ is uncountable then we have an example where the closure of a compact set is not a Lindelöf space!.

All these examples and more can be found presented with more detail in the wonderful book, Counterexamples in Topology by Lynn Arthur Steen and J. Arthur Seebach Jr. I strongly advise if you're interested in point set topology, that you get a copy.

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    $\begingroup$ I don't think this really answers the question - the OP is asking why they should care about non-$T_1$ spaces, not for some basic examples. If you add a little about why e.g. the Alexandrov topology on a poset is interesting, I think that would be closer to an answer. (Similarly, while I love Steen/Seebach, it doesn't really explain why for example non-$T_1$ spaces are useful, it just gives a bunch of (really really cool) examples.) $\endgroup$ – Noah Schweber Jun 10 '16 at 3:49
  • $\begingroup$ @NoahSchweber I don't agree. I think it perfectly answers the question. "Why should we study this object?" "Well, there are important examples of this object". I think that is perfectly good justification. $\endgroup$ – user223391 Jun 10 '16 at 3:53
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    $\begingroup$ @ZacharySelk But the answer doesn't say anything about why these examples are important . . . $\endgroup$ – Noah Schweber Jun 10 '16 at 3:54
  • $\begingroup$ @NoahSchweber Okay, I get your point. $\endgroup$ – user223391 Jun 10 '16 at 3:55
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    $\begingroup$ @Mathemagician1234 This still doesn't really explain why we should care about these - if I were a mathematician with no intrinsic interest in point-set topology, this wouldn't convince me to pay attention to non-$T_1$ spaces. (I am interested in non-$T_1$ spaces, by the way.) $\endgroup$ – Noah Schweber Jun 10 '16 at 4:10

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