Suppose that I have a function that outputs the concatenation of its inputs $f(a,b)=a.b$ .This can be represented with a topological space $(X,\tau)$ where $X=\{a,b\}$ and $\tau=\{\emptyset,X\}$. However, the topology only describes that there exist an open set $\{a,b\}$ that comprises $a$ and $b$ (i.e. where $a\in X$ and $b\in X$) but doesn't describe the order within the concatenation $a.b$ (i.e. that $a$ comes before $b$). So, how can I change/modify/extend the topological space so that I can express that $a<_{a.b}b$.
SOLUTION - "Dana Scott - Outline of a Mathematical Theory of Computation" (1970) (pdf here: https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf)
I will assume that the function $f$ that you are defining has domain and codomain $X^2$ for $X = \{a,b\}$, otherwise we wouldn't be talking about a well-defined object; moreover, as I mention in the comments, note that the function $f : X^2 \to X^2$ as you've defined it is just the identity on $X^2$.
Having established this, here is a simple way of coding the "order of concatenation" given by $f$ topologically. In this example there are in fact $4$ possible ways of concatenating the only $2$ elements of $X$, each corresponding to the $4$ elements of $X^2$ (you haven't mentioned that the two arguments in $f$ must be distinct!); after pointing this out, we let $\pi_1, \pi_2 :X^2 \to X$ be the projection maps onto the first and second coordinate respectively and by setting $\tau = \{ \varnothing, \{a\}, X\}$ to be our topology on $X$ we have that
$$a <_{a.b} b \iff \{\pi_1(f(x,y))\} \text{ is open and }\{\pi_2(f(x,y))\} \text{ is closed}$$ $$b <_{b.a} a \iff \{\pi_1(f(x,y))\} \text{ is closed and }\{\pi_2(f(x,y))\} \text{ is open}$$ $$a =_{a.a} a \iff \{\pi_1(f(x,y))\} \text{ and }\{\pi_2(f(x,y))\} \text{ are open}$$ $$b =_{b.b} b \iff \{\pi_1(f(x,y))\} \text{ and }\{\pi_2(f(x,y))\} \text{ are closed}$$