Why study non-T1 topological spaces?

Solution 1:

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.

Solution 2:

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)

Solution 3:

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.