Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants.
Topology has sometimes been called rubber-sheet geometry, because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing).
In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
Points are used in the basic language of geometry, physics, vector graphics (both 2d and 3d), and many other fields.
Euclid's axioms or postulates assert in some cases that points exist: for example, they assert that if two lines on a plane are not parallel, there is exactly one point that lies on both of them.
The description of a point is quite similar to the description of a spatial vector, which also can exist in space with dimensions from one to many.
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