Lower limit topology
In mathematics, the lower limit topology is a topology defined on the real numbers R which has a number of interesting properties. The topology is that generated by a basis of all half-open intervals [a,b), where a and b are real numbers. It is also known as the right half-open interval topology.The lower limit topology is finer, or a superset, of the standard topology on the real numbers (which is generated by open intervals). Although its structure is relatively simple, it is still, like the Cantor set and the long line, often a useful counterexample.
The resulting topological space S, sometimes written 
, has a number of interesting properties:
- For any real a and b, the interval [a, b) are both open and closed, or clopen.
- For all real a, the sets {x
R | x 
a} and {x R | x > a} are also clopen.
- S is Hausdorff and first countable but not second countable (and hence not metrizable).
- S is Lindelöf, regular, separable, and paracompact but not sigma-compact or locally compact.
