What is a point-split? I encountered the term point-split [1] several times and would like to know what this concept is all about. 
From my understanding, a point is splitted by adding $ε$ and $-ε$ to a local point $x$ with $ε → 0$.


*

*What is a point-split?

*Why the blur of locality?

*Is it a general concept at all? 

*If so, why was it introduced? 

*What is it used for? 

*Does it belong to topology?


Grateful for your answers.
[1] https://arxiv.org/abs/hep-th/9803244
 A: *

*A point-splitting procedure is one way to make sense of composite fields in QFT. As an easy example, take a free scalar field $\phi(x)$ in Euclidean signature, in $d$ dimensions. Consider the problem of making sense of the local square $\phi(x)^2$ "inside correlations". The point-splitting amounts to doing so via the limit
$$
\phi(x)^2:=\lim_{\epsilon\rightarrow 0}\ \ 
[\ \phi(x+\epsilon)\phi(x-\epsilon)-\langle\phi(x+\epsilon)\phi(x-\epsilon)\rangle\ ]\ .
$$

*Indeed this seems to break locality but one recovers it because $\epsilon$ is sent to $0$. The reason one has to do something like is that correlations diverge at coinciding points. For example
$$
\langle\phi(x)\phi(y)\rangle\sim \frac{1}{|x-y|^{d-2}}
$$
so you need to subtract a divergent quantity before taking the limit. You also need to introduce the splitting $\epsilon$ to see what you need to subtract.

*Yes. The basic idea of this point-splitting procedure was introduced by Dirac, Heisenberg, Euler and Valatin (see the references and discussion of the history of this method in Section 1.11 of my article "A Second-Quantized Kolmogorov-Chentsov Theorem"). However, the general framework for understanding this is Wilson's Operator Product Expansion.

*Already answered in 2.

*Already answered in 2.

*Not really. This is (hard) analysis, not topology. Although, I should mention that similar constructions appear in compactifications of configuration spaces which topologists and algebraic geometers use. I am not an expert on this, but you can look up "factorization algebras", "Fulton-MacPherson compactification", "wonderful compactifications" for more on this aspect.
