Constructing an Indiscrete Topological Space: Identical Open and Closed Sets

Understanding the concept of indiscrete topological spaces and how their open and closed sets are identical is crucial for advanced topics in topology. This article will guide you through the construction of an indiscrete topological space, explore its properties, and provide additional insights related to partitions and equivalence relations.

Definition of an Indiscrete Topological Space

To construct an indiscrete topological space, follow these steps:

Choose a Set: Let X be any non-empty set. For example, you can choose X {a, b}. Define the Topology: The indiscrete topology on X consists of only two sets: The empty set emptyset The entire set X itself Therefore, the topology tau can be defined as: (tau { emptyset, X })

Properties of the Indiscrete Topology

Open Sets

The only open sets in this topology are emptyset and X.

Closed Sets

The closed sets are defined as the complements of the open sets. Since the complements of emptyset and X are X and emptyset respectively, the closed sets are also { emptyset, X }.

Conclusion

In an indiscrete topological space, the open sets and closed sets are indeed identical as both consist of just emptyset and X. This construction can be applied to any non-empty set, making it a very general example of an indiscrete topology.

The indiscrete topology is the one where only the whole space and the empty set are open. It is important to note that if the open sets are identical to the closed sets, the intersection of any arbitrary family of open sets is open. Let's explore this further using the example of a topological space defined by a partition.

Partitions and Indiscrete Topologies

Consider any non-empty set X and fix a partition mathcal{F} of X such that mathcal{F} is a family of pairwise disjoint non-empty subsets of X whose union recovers the entire X. If you consider mathcal{F} as a basis for a topology on X, this topology will have the property that the family of open sets and the family of closed sets coincide since the open sets will be unions of subfamilies of the partition, and so will the closed sets.

This construction can be understood as being similar to the indiscrete topology, where the only open sets are the whole space and the empty set. However, it is equivalent to forming the topological sum of all parts of the partition, each of them endowed with the indiscrete topology.

Implications and Equivalence Relations

If the open sets are identical to the closed sets, the intersection of any arbitrary family of open sets is open. In this topology, consider the intersection (U_p) of the open sets containing a single point (p). This set (U_p) is open. If (q) is in (U_p), then because (U_p) is an open set containing (q), we have that (U_q) is a subset of (U_p). It cannot be a proper subset containing (p), else (U_p) would already contain it. Also, it cannot be a proper subset of (U_p) not containing (p), because its complement would be open and contain (p), leading a contradiction since (U_p) would then be contained in the complement of (U_q).

Hence, (U_p) is a set of equivalence classes on the space, and the open sets are the sets respecting the equivalence relation. As long as the equivalence relation is not the same as equality, this is not the discrete topology but a discrete topology on the quotient set pulled back to the original set.