Lower Limit Topology

In the lower limit topology, an open set is defined as any union of left-closed, right-open intervals of the form [a, b), where a < b.

In essence, an interval is open in the lower limit topology if it includes its lower bound but not its upper bound.

The foundation of this topology is described as follows:

$$ B = \{ [a,b) ⊂ R \ | \ a<b \} $$

Each element of this foundation features a lower bound that is included within the subset itself.

Note: This is a unique topology defined on the set of real numbers (R) that stands in contrast to the standard topology on the real numbers (R), where open intervals are of the form (a, b) and exclude both ends.

The lower limit topology is often used as an example in topology courses to demonstrate how the choice of topology affects the definition of open sets.

In this topology, left-closed, right-open intervals [a,b) are considered open sets.

A Practical Example

A practical example of the lower limit topology can be shown by considering the set of real numbers R with left-closed, right-open intervals as open sets.

For instance, the collection of subsets like [0,2), [1,4), [-4,2), etc.

The set of all these left-closed, right-open intervals forms the foundation of the lower limit topology.

And so forth.