A topological space is a set $ X $ , known as the underlying set, together with a topology T of $ X $ that assigns to each element in $ X $ a neighborhood consisting of a subset of $ X $.
Topology has sometimes been called rubber-sheet geometry, because in topology of 2 dimensions, there is no difference between a circle and a square (a circle made out of a rubber band can be stretched into a square) but there is a difference between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing). The spaces studied in topology are called topological spaces.
Euler's formula is an example of something that can be proven using only topology. It remains true even if the rubber sheet is stretched.
This article incorporates text from Topology