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Overlapping interval topology

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Not to be confused with Interlocking 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 ] {\displaystyle } of the real number line, the open sets of the topology are generated from the half-open intervals ( a , 1 ] {\displaystyle (a,1]} with a < 0 {\displaystyle a<0} and [ 1 , b ) {\displaystyle [-1,b)} with b > 0 {\displaystyle b>0} . The topology therefore consists of intervals of the form [ 1 , b ) {\displaystyle [-1,b)} , ( a , b ) {\displaystyle (a,b)} , and ( a , 1 ] {\displaystyle (a,1]} with a < 0 < b {\displaystyle a<0<b} , together with [ 1 , 1 ] {\displaystyle } itself and the empty set.

Properties

Any two distinct points in [ 1 , 1 ] {\displaystyle } 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 ] {\displaystyle } , making [ 1 , 1 ] {\displaystyle } 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 ) {\displaystyle [-1,s)} , ( r , s ) {\displaystyle (r,s)} and ( r , 1 ] {\displaystyle (r,1]} with r < 0 < s {\displaystyle r<0<s} and r and s rational.

See also

References

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