Given the $7$-element set $A = \{ a ,b,c,d,e,f,g \}$, find a collection $T$ of $3$-element subsets of $A$ such that each pair of elements from $A$ occurs exactly once on one of the subsets of $T$.
To solve this problem, we need to find a collection \( T \) of 3-element subsets of \( A \) such that each pair of elements from \( A \) occurs exactly once in one of the subsets of \( T \). This is a classic problem in combinatorial design theory, specifically a Balanced Incomplete Block Design (BIBD) with parameters \( (7, 3, 1) \).

1. **Understanding the Parameters**:
   - We have a set \( A \) with 7 elements.
   - We need subsets (blocks) of size 3.
   - Each pair of elements from \( A \) should appear exactly once in one of the subsets.

2. **Fano Plane**:
   - The Fano plane is the smallest finite projective plane, and it is a BIBD with parameters \( (7, 3, 1) \).
   - It consists of 7 points and 7 lines (subsets), where each line contains exactly 3 points, and each pair of points is contained in exactly one line.

3. **Constructing the Subsets**:
   - We can label the elements of \( A \) as \( \{a, b, c, d, e, f, g\} \).
   - The Fano plane can be represented by the following 3-element subsets:
     \[
     \{a, b, c\}, \{a, d, e\}, \{a, f, g\}, \{b, d, f\}, \{b, e, g\}, \{c, d, g\}, \{c, e, f\}
     \]

4. **Verification**:
   - We need to verify that each pair of elements from \( A \) appears exactly once in one of the subsets.
   - List all pairs and check their occurrence:
     - \( \{a, b\} \) in \( \{a, b, c\} \)
     - \( \{a, c\} \) in \( \{a, b, c\} \)
     - \( \{a, d\} \) in \( \{a, d, e\} \)
     - \( \{a, e\} \) in \( \{a, d, e\} \)
     - \( \{a, f\} \) in \( \{a, f, g\} \)
     - \( \{a, g\} \) in \( \{a, f, g\} \)
     - \( \{b, c\} \) in \( \{a, b, c\} \)
     - \( \{b, d\} \) in \( \{b, d, f\} \)
     - \( \{b, e\} \) in \( \{b, e, g\} \)
     - \( \{b, f\} \) in \( \{b, d, f\} \)
     - \( \{b, g\} \) in \( \{b, e, g\} \)
     - \( \{c, d\} \) in \( \{c, d, g\} \)
     - \( \{c, e\} \) in \( \{c, e, f\} \)
     - \( \{c, f\} \) in \( \{c, e, f\} \)
     - \( \{c, g\} \) in \( \{c, d, g\} \)
     - \( \{d, e\} \) in \( \{a, d, e\} \)
     - \( \{d, f\} \) in \( \{b, d, f\} \)
     - \( \{d, g\} \) in \( \{c, d, g\} \)
     - \( \{e, f\} \) in \( \{c, e, f\} \)
     - \( \{e, g\} \) in \( \{b, e, g\} \)
     - \( \{f, g\} \) in \( \{a, f, g\} \)

   - Each pair appears exactly once, confirming the correctness of the subsets.

The final answer is \( \boxed{ T = \{\{a, b, c\}, \{a, d, e\}, \{a, f, g\}, \{b, d, f\}, \{b, e, g\}, \{c, d, g\}, \{c, e, f\}\} } \).