Upper Limit Topology

In upper limit topology, an open set is defined as any union of right semi-open intervals of the form (a, b], where a < b.

This definition clarifies that, for an interval to be considered open in upper limit topology, it must include its upper bound but exclude its lower bound.

Formally, the basis for this topology is expressed as:

$$ B = \{ (a,b] \subset R \ | \ a<b \} $$

This foundational structure signifies that each element is characterized by the inclusion of the upper bound within the subset.

Note. It's interesting to observe the duality with the lower limit topology, where open intervals are of the form [a,b) and include the lower bound. This comparison highlights how the choice of topology significantly impacts the very definition of set openness.

The upper limit topology offers a vital perspective in the field of topology, illustrating how variations in basic definitions can lead to differing properties and outcomes.

A Practical Example

Consider the set of real numbers R with right semi-open intervals as open sets.

Examples of such sets include (1,3], (2,6], (−3,5], and so on.

The collection of all these semi-closed intervals forms the basis of the upper limit topology.

In each interval, the upper bound is included in the set, while the lower bound is excluded.

And so forth.