Trivial Topology
The trivial (or minimal) topology on a set X is defined by just two elements: the empty set and the set itself. $$ T = \{ \emptyset , X \} $$
This is known as the trivial topology, representing the most basic topological structure that can be applied to a set.
Such a topology is comprised exclusively of the empty set Ø and the set X, effectively including only the improper subsets of X.
Understanding the Basics
When we take a non-empty set X and assign it a trivial topology T, we're setting up a fundamental structure.
$$ (X, T) $$
Here, the trivial topology T is simply a set with two components: the empty set () and the set X itself.
$$ T = \{ \emptyset , X \} $$
Opting for these particular sets is crucial as it ensures compliance with topological conditions.
To qualify as a topology on X, T must meet three essential criteria:
- Both the empty set Ø and the full set X are included in T.
- Any union of open sets within T remains an open set in T.
- The intersection of any two open sets in T is also an open set in T.
For the topology T={Ø, X}, these requirements are naturally fulfilled.
Proof. By definition, the empty set and X are already part of T.
Where X is an open set by our initial assumption, while the empty set is universally accepted as open in topology.
Additionally, with no other sets present in T, neither unions nor intersections can breach topological rules.
Therefore, all necessary topological conditions are satisfied.
Why is it Called the Minimal Topology?
The trivial topology is dubbed the minimal topology due to its status as the simplest conceivable topological structure on set X.
A topology is considered minimal if removing any element from T would mean it no longer defines a topology.
This stipulation arises from the fundamental requirement that a topology on set X must include at least the empty set Ø and the set X itself.
Given that the trivial topology T={Ø,X} only contains these two elements, it leaves no room for removal.
Should we extract either the empty set Ø or X from T, the set would fail to fulfill the basic criteria of a topology.
Thus, the trivial topology T={Ø, X} represents the most streamlined or minimal topology achievable on X.
Note. While the trivial topology is an elegantly simple construct, useful for theoretical exploration, it's seldom seen in practice due to its lack of structural complexity or insights about the defined set. It serves as a minimal extreme among the spectrum of topologies that could be devised for a set. In contrast, the discrete topology, considering every subset of X as open, stands at the opposite end of this spectrum.
And so forth
