Temporal topology involves non-metric temporal relationships. Time is fundamentally different from space in that all temporal entities can be mapped onto a one-dimensional, one-directional line. The topology along the time line is more complicated than that of points and arcs along a spatial line. With time the start and end of a duration take on great significance, as they indicate when the object in question came into existence or ceased to exist. (This work is based in part of on that of Pierre Gagnon and Yvan B‚dard at Laval.) temporallyDisjoint: No part of A is simultaneous with any part of B. temporallyIntersect: The compliment of temporallyDisjoint, i.e., not temporally disjoint. The cases described below are the detailed ways in which two temporal objects may intersect temporally. atStart: A occurs at a point in time simultaneous with the start of the duration of B. (A is a point; B is a duration.) atEnd: A occurs at a point in time simultaneous with the end of the duration of B. (A is a point; B is a duration.) follows: A begins when B ends. (A and B are both durations.) overlapsAtStart: The start of A does not intersect B, and the end of A intersects B, but not at its start or end, and the start of B intersects A, but not at its start or end. (A and B are both durations.) overlapsAtEnd: The end of A does not intersect B, and
the start of A intersects B, but not at its start or end,
and the end of B intersects A, but not at its start or end.
(A and B are both durations.)
during: All of A is simultaneous with some part of B, excluding the start and end of B.
(A is a point or duration, and B is a duration.)
duringFromStart: All of A is simultaneous with some part of B, excluding the end of B, and
the start of A is simultaneous with the start of B.
(A and B are both durations.)
duringToEnd: All of A is simultaneous with some part of B, excluding the start of B, and
the end of A is simultaneous with the end of B.
(A and B are both durations.)
simultaneous: A in its entirety and B in its entirety occur at exactly the same time, i.e., they occupy exactly the same portion of the time line.
(A and B are both points or both durations.)
These terms are defined rigorously using set theory constructs, based on boundary (start and end) and interior (interval) comparisons.
If object A is a temporal object mappable onto a time line, then its birth (start) and death (end) are indicated as As and Ae, respectively. The interval between the start and end is designated as Ai. A point in time is considered to have no boundaries, i.e., no start and no end; it does have an interval (an interior), corresponding to the situation of a point in space. These conditions are described in the following table.
To define topologic relations, a number of interior and boundary comparisons can be made. This is similar to what was done spatially; however, in this case the comparisons are simpler because the geometry of the time line is simpler. All the comparisons needed are given on the table on the right. (Specifically, the concepts of partial and total - the two right most columns on the corresponding spatial table - are addressed implicitly by various combinations of the nine boolean values in the temporal table.) To handle all possible temporal topological relationships, all nine comparisons must be made. The high levels relationships (atStart, atEnd, etc.) listed earlier are defined on the table which follows.
(Temporally disjoint implies either before or after. To distinguish one condition from the other cannot be done topologically. The temporal relationship Precedence addresses these comparisons.)