The "topological structure" of the tuples of images and preimages of a function

Question

Let $A$ and $B$ be two spaces. And let $f:A\to B$. Assume for now that $f$ is a bijection.

Then for every $a\subseteq A$, we can take the image of $a$: $f(a)=b$ for some $b\subseteq B$.

Now take the tuple $(a,b)$ (the tuple of the $a$ and its image $b$, or alternatively, $b$ and its preimage $a$). Denote by $T$ the set of all tuples $(a,b)$ such that $f(a)=b$.

$T$ seems to have a "topology-like structure":

1. $(\emptyset, \emptyset)\in T$, $(A,B)\in T$

2. If $(a_i,b_i)\in T$ for $i\in I$, then $(\bigcup _ia_i,\bigcup _ib_i)\in T$

3. If $(a_i,b_i)\in T$ for $i\in \mathbb N$, then $(\bigcap _ia_i,\bigcap _ib_i)\in T$

However, this is not literally a topology, since the "topology-like" structure is over the elements of the tuples, not over the tuples themselves.

1. Is there a name for this "topology-like structure"

2. Can we reformulate this to make it into an actual topology?

Answer 1

Closest to what you are considering are Induced Topologies. If one of the spaces is equipped with a topology, then you may equip the other with a corresponding induced topology and consequently the product space $A\times B$ carries a product topology.

One natural example in your setting would be to equip $B$ with the discrete topology (every point is its own neighborhood) and then consider the induced topology (via $f$) on $A$. This topology then considers point, which map to the same element as 'close to each other'.