The word “topology” is derived from the
Greek word “τοπος,”
which means “position” or “location.”
A simplified and thus partial definition has often been
used (Croom, 1989, page 2): “topology deals with
geometric properties which are dependent only upon the
relative positions of the components of figures and not
upon such concepts as length, size, and magnitude.”
Topology deals with those properties of an object that
remain invariant under continuous transformations (specifically
bending, stretching, and squeezing, but not breaking or
tearing). Topological notions and methods have illuminated
and clarified basic structural concepts in diverse branches
of modern mathematics. However, the influence of topology
extends to almost every other discipline that formerly
was not considered amenable to mathematical handling. For
example, topology supports design and representation of
mechanical devices, communication and transportation networks,
topographic maps, and planning and controlling of complex
activities. In addition, aspects of topology are closely
related to symbolic logic, which forms the foundation of
artificial intelligence. In the same way that the Euclidean
plane satisfies certain axioms or postulates, it can be
shown that certain abstract spaces—where the relations
of points to sets and continuity of functions are important—have
definite properties that can be analyzed without examining
these spaces individually. By approaching engineering design
from this abstract point of view, it is possible to use
topological methods to study collections of geometric objects
or collections of entities that are of concern in design
analysis or synthesis. These collections of objects and
or entities can be treated as spaces, and the elements in them as points.
On some operations on soft topological spaces
