Topology (from the Greek τόπος, "place", and λόγος, "study") is the mathematical study of shapes and topological spaces. It is an area of mathematics concerned with the:
properties of space that are preserved under continuous deformations including stretching and bending, but not tearing or gluing.
This includes such properties as connectedness, continuity and boundary.
A doughnut (torus) is said to be topologically equivalent to a mug. image: wikipedia
Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.
Such ideas go back to Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place").
Origin of the term topology
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.
Topology has many subfields
General topology
establishes the foundational aspects of topology and investigates properties of topological spaces and investigates concepts inherent to topological spaces. It includes point-set topology, which is the foundational topology used in all other branches (including topics like compactness and connectedness).
Algebraic topology
tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups.
Differential topology
is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.
Geometric topology
primarily studies manifolds and their embeddings (placements) in other manifolds. A particularly active area is low dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots.
A three-dimensional depiction of a thickened trefoil knot, the simplest non-trivial knot. image: wikipedia
source: http://en.wikipedia.org/w/index.php?title=Topology&oldid=607905774