Overlapping Interval Topology

In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.

Definition

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

Properties

Any two distinct points in 1,1/math> are Distinct (mathematics)">distinct points in 1,1/math> 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/math>, making 1,1/math> with the overlapping interval topology an example of a T₀ space that is not a T1 space">T₁ space.

The overlapping interval topology is second countable">T1_space.html" ;"title="T0 space">T₀ space that is not a T1 space">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 $r < 0 < s$ and ''r'' and ''s'' rational.

See also

* 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

References

* {{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 , origyear=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)''

Topological spaces