In mathematics, more specifically general topology, the nested interval topology is an example of a topology given to the open interval (0,1), i.e. the set of all real numbers x such that 0 < x < 1. The open interval (0,1) is the set of all real numbers between 0 and 1; but not including either 0 or 1.
To give the set (0,1) a topology means to say which subsets of (0,1) are "open", and to do so in a way that the following axioms are met:[1]
The union of open sets is an open set.
The finite intersection of open sets is an open set.
The set (0,1) and the empty set ∅ are open sets.
Construction
The set (0,1) and the empty set ∅ are required to be open sets, and so we define (0,1) and ∅ to be open sets in this topology. The other open sets in this topology are all of the form (0,1 − 1/n) where n is a positive whole number greater than or equal to two i.e. n = 2, 3, 4, 5, ….[1]
Properties
The nested interval topology is neither Hausdorff nor T1. In fact, if x is an element of (0,1), then the closure of the singleton set {x} is the half-open interval [1 − 1/n,1), where n is maximal such that n ≤ (1 − x)−1.[1]
The nested interval topology is not compact. It is, however, strongly Lindelöf since there are only countably many open sets.[1]
The nested interval topology is hyperconnected and hence connected.[1]
The nested interval topology is Alexandrov.[1]
References
Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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