Nested interval topology
In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval, i.e. the set of all real numbers x such that. The open interval is the set of all real numbers between 0 and 1; but not including either 0 or 1.
To give the set a topology means to say which subsets of are "open", and to do so in a way that the following axioms are met:
- The union of open sets is an open set.
- The finite intersection of open sets is an open set.
- The set and the empty set ∅ are open sets.
Construction
Properties
- The nested interval topology is neither Hausdorff nor T1. In fact, if x is an element of, then the closure of the singleton set is the half-open interval, where n is maximal such that.
- The nested interval topology is not compact. It is, however, strongly Lindelöf since there are only countably many open sets.
- The nested interval topology is hyperconnected and hence connected.
- The nested interval topology is Alexandrov.