Topology is a major area of mathematics concerned with spatial properties
that are preserved under continuous deformations of objects, for example,
deformations that involve stretching, but no tearing or gluing. It emerged
through the development of concepts from geometry and set theory, such
as space, dimension, and transformation.
Ideas that are now classified as topological were expressed as early as
1736. Toward the end of the 19th century, a distinct discipline developed,
which was referred to in Latin as the geometria situs («geometry of place»)
or analysis situs (Greek-Latin for «picking apart of place»). This later
acquired the modern name of topology. By the middle of the 20th century,
topology had become an important area of study within mathematics.
The word topology is used both for the mathematical discipline and for a
family of sets with certain properties that are used to define a topological
space, a basic object of topology. Of particular importance are
homeomorphisms, which can be defined as continuous functions with
a continuous inverse. For instance, the function y = x3 is a homeomorphism
of the real line.
Topology includes many subfields. The most basic and traditional division
within topology is point-set topology, which establishes the foundational
aspects of topology and investigates concepts inherent to topological spaces
(basic examples include compactness and connectedness); algebraic topology,
which generally tries to measure degrees of connectivity using algebraic
constructs such as homotopy groups and homology; and geometric topology,
which primarily studies manifolds and their embeddings (placements) in other
manifolds. Some of the most active areas, such as low dimensional topology
and graph theory, do not fit neatly in this division.