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Topology

Topology is a branch of mathematics studying the properties of figures in space under some specific transformations. It is sometimes called “rubber geometry”, since it deals with the way geometrical figures placed on an imaginary “rubber” can be deformed when this rubber is stretched or bent. What is key in topology is the continuity of all transformations like these, i.e., the “rubber” might not be torn or glued.

One of key questions in topology is asking, under which circumstances one geometrical figure can be continuously transformed into another figure in a way described above. If it is possible, these figures are considered topologically the same (or homotop). Figures, which are not homotop, are in some fundamental way different (because they cannot be transformed into each other in a smooth way).

Trying to find out, if two figures are homotop, topologists developed different powerful methods. One of these methods is so fundamental that it is now considered one of the most important methods not only in topology but also in mathematics as a whole – the concept of invariants. Invariants are unchanging properties of a mathematical object (in topology, these objects are geometrical figures), no matter how strongly it is deformed.

Invariants are so powerful because they allow for a quick decision for two figures being homotop or not, even if they look apparently not very similar, without trying out all possible ways to deform one figure into another (provide an example).
It is usually very easy to prove that two figures are definitely not homotop if the values of some invariant for both figures are different. If they are equal, however, it is much harder to prove, that they are homotop!

Invariants are also so powerful because they help to classify all figures into so-called homotopy classes, i.e. in which figures are considered to be “essentially the same”. Finding all existing homotopy classes in different spaces is another important question of topology and a field of ongoing research.