Final answer:
The elements of a topology are called open sets because they align with the open interval concept in analysis, which has no boundary points. The whole set X is always open in a topological space by definition to ensure consistency in defining properties like continuity. Including the whole set as open does not conflict with being without boundary; rather it lacks a boundary within the space.
Step-by-step explanation:
The elements of a topology are called open sets because this concept aligns with the notion of an "open interval" in analysis, where no boundary points are included. An open set is a fundamental concept in topology because it helps define a topological space, which is a set equipped with a collection of open subsets that satisfy certain axioms (such as any union and finite intersection of open sets is open).
In every topology, the whole set X is considered an open set by definition. This is because the axioms of a topology require the entire space and the empty set to be included as open sets. Including the whole set as an open set ensures that certain properties, like continuity, can be defined in a consistent way across all spaces. The whole set X contains every point by its very nature, but being open, in this case, does not conflict with the intuitive idea of "without boundary" because the set does not exclude anything and hence lacks a boundary within the context of the space.