Point Set Topology

Law of the Excluded Middle

The Law of the Excluded Middle is one of the three Laws of Thought. In point-set topology, the law of excluded middle is a fundamental principle that comes from classical logic. This law states that for any proposition ( P ), either ( P ) is true or its negation ( \neg P ) is true. In the context of topology, this principle is often implicitly assumed when dealing with open and closed sets.

For example, in classical point-set topology, a set ( U ) in a topological space ( X ) is defined as open if its complement ( X ∖ U ) is closed, and vice versa. This relies on the law of excluded middle because it assumes that every point in ( X ) either belongs to ( U ) or to its complement ( X ∖ U ), with no middle ground1.

However, it’s worth noting that in constructive mathematics or intuitionistic logic, the law of excluded middle is not accepted. This leads to different approaches and definitions in topology, where the existence of certain sets or points might not be as straightforward2.