Here is a quote from John Lee's Introduction to Topological Manifolds, Second Edition:
The definition of topological spaces is wonderfully flexible, and can be used to describe a rich assortment of concepts of "space". However, without further qualification, arbitrary topological spaces are far too general for most purposes, because they include some spaces whose behavior contradicts many of our basic spatial intuitions.
By requiring a space to be Hausdorff we exclude "pathological" topological space, such as the one in which a sequence could converge to more than one point.
Then, why do we not require all topological spaces to have the Hausdorff property? What do we gain by defining topological spaces to be more general spaces than Hausdorff spaces?
Best Answer
In fact, in the early 20th century, when topology was being "invented"or "discovered" as a separate mathematical branch, there were several approaches in defining topologies in the first place, as a set with some extra structure. Kuratowski used axioms for closure (as did Čech) and also demanded that $\overline{\{x\}} = \{x\}$ in some of his texts (so $T_1$ was assumed throughout). Fréchet used convergence notions (of sequences) and also assumed that constant sequences had unique limits (so $T_1$ too). Hausdorff used an axiom system based on neighbourhood systems and assumed as one of the axioms that two distinct points had at least 2 disjoint respective neighbourhoods, the axiom that was later named after him. So in the early days people "sneaked in" low separation axioms as part of their definitions, mostly for convenience. Later the open sets/closed sets axioms developed and were quite generally found to be convenient (and similar to other structures being developed at that time, like $\sigma$-algebras etc.) and people proved the general "equivalence" of many of these approaches. At that time the separation axioms were formulated as separate assumptions.
The system with bare-bones axioms and simple extra assumptions (Trennungsaxiome like $T_0, T_1, T_2$ etc.) won out. That way we can easily tell which results need which extra assumptions etc.
And of course later many applications of non-Hausdorff spaces were found too, which helps.