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This problem is from the field of topology, which studies how spaces are structured and how they behave under continuous deformations. Let's break down both parts of the question:

### Part a) Differentiating between a finest topology and a coarsest topology

- **Finest Topology**: The finest topology on a set \( X \) is the discrete topology, where every subset of \( X \) is considered open. This topology provides the greatest amount of detail, as it allows every element to be its own open set.

- **Coarsest Topology**: The coarsest topology is the indiscrete topology, also known as the trivial topology. In this case, only the entire set \( X \) and the empty set \( \emptyset \) are open. This is the least detailed topology because it has the fewest open sets possible.

**Key Point**: The finest topology gives the greatest flexibility in terms of open sets, while the coarsest topology is the most restrictive.

### Part b) Finding the indiscrete and discrete topologies on \( X = \{a, b\} \)

- **Indiscrete Topology on \( X = \{a, b\} \)**: 
  The indiscrete topology on \( X \) only includes two open sets:
  1. The empty set \( \emptyset \)
  2. The whole set \( X = \{a, b\} \)

- **Discrete Topology on \( X = \{a, b\} \)**: 
  The discrete topology on \( X \) includes all possible subsets of \( X \). These are:
  1. \( \emptyset \)
  2. \( \{a\} \)
  3. \( \{b\} \)
  4. \( \{a, b\} \)

To compare the two:
- The indiscrete topology is **coarser** than the discrete topology, meaning it has fewer open sets. The discrete topology is **finer** because it has more open sets, allowing for more granularity in terms of what can be considered "open."

### Conclusion:
- The indiscrete topology is coarser, and the discrete topology is finer.
  
Would you like more details or have further questions about these concepts? Here are five related questions to deepen your understanding:

1. What are examples of topologies that are neither discrete nor indiscrete?
2. How do continuous functions behave between spaces with discrete and indiscrete topologies?
3. Can there be more than one finest or coarsest topology on a set?
4. How do we define a basis for a topology, and how does it apply to discrete and indiscrete cases?
5. How does the concept of open sets in a topology relate to limits in calculus?

**Tip**: When studying topologies, always think about how open sets interact and whether they give a detailed or coarse view of the space's structure.

Let X be a non-empty set. a) Differentiate between a finest topology and coarsest topology on X. b) Let X = {a, b}. Find the indiscrete and discrete topologies on X, and hence show one is coarser than the other.

Explore the differences between the finest and coarsest topologies on a set X. Learn how to find both indiscrete and discrete topologies for X = {a, b} and understand why one is coarser than the other. Ideal for undergraduate students in topology.

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