Overlapping interval topology

Given the closed interval $[-1,1]$ of the real number line, the open sets of the topology are generated from the half-open intervals $(a,1]$ with and $[-1,b)$ with $b\; >\; 0$. The topology therefore consists of intervals of the form $[-1,b)$, $(a,b)$, and $(a,1]$ with , together with $[-1,1]$ itself and the empty set.

Any two distinct points in $[-1,1]$ are topologically distinguishable under the overlapping interval topology as one can always find an open set containing one but not the other point. However, every non-empty open set contains the point 0 which can therefore not be separated from any other point in $[-1,1]$, making $[-1,1]$ with the overlapping interval topology an example of a T₀ space that is not a T₁ space.

The overlapping interval topology is second countable, with a countable basis being given by the intervals $[-1,s)$, $(r,s)$ and $(r,1]$ with and r and s rational.

• List of topologies
• Particular point topology, a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space

• {{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=Counterexamples in Topology | orig-date=1978 | publisher=Springer-Verlag | location=Berlin, New York | edition=Dover reprint of 1978 | isbn=978-0-486-68735-3 |mr=507446 | year=1995}} (See example 53)

Category:Topological spaces