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[edit]

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

Properties[edit]

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 r < 0 < s and r and s rational (and thus countable).

References[edit]