What is Topology?

There are various views on what is and what is not "Topology". Even trickier, there are various quite distinct areas of topology. MathWorld introduces topology as follows:

Topology is the mathematical study of the properties that are preserved through deformations, twistings, and stretchings of objects. Tearing, however, is not allowed.

Weisstein, Eric W. "Topology." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Topology.html

As far as this goes, everyone can agree to it. But the differences start when trying to formalize what is meant by "deformations, twistings, and stretchings" and what kind of "objects" should be considered.

Analytic topology, often referred to as "general topology" or "point-set topology", takes one of the broadest views on these issues. On the one hand this leads to widely applicable, powerful and, in the author's opinion, elegant results. But this broad view also leads to a rather large number of counterexamples to conjectures, which many mathematicians would like to be true.

A thorough discussion of some of these issues and some excellent introductory material can be found at What is Topology? on the Topology Atlas.

The term 'Topologie' itself was apparently introduced in the mid 18th century. A thorough introduction to the fascinating history of the subject can be found at A history of Topology

Johann Benedict Listing (1802-1882) was the first to use the word topology. Listing's topological ideas were due mainly to Gauss, although Gauss himself chose not to publish any work on topology. Listing wrote a paper in 1847 called Vorstudien zur Topologie although he had already used the word for ten years in correspondence. The 1847 paper is not very important, although he also introduces the idea of a complex, since it is extremely elementary. In 1861 Listing published a much more important paper in which he described the Möbius band (4 years before Möbius) and studied components of surfaces and connectivity.

J J O'Connor and E F Robertson from A history of Topology