Abstract:
I am going to introduce some properties of coarse structure. Coarse
space is defined for large scale in metric space similar to the tools provided by
topology for analyzing behavior at small distance, as topological property can
be defined entirely in terms of open sets. Analogously a large scale property
can be defined entirely in terms of controlled sets. The properties we required
were that the maps were coarse (proper and bornologous), but why do these
maps imply that the spaces have the same large structure? Essentially this has
to do with contractibility. Spaces which are the same on a large scale can be
scaled so that the points are not too far away from each other, but we are not
concerned with any differences on small scale that may arise. In addition I am
going to explain some basic definition related with the title of my research work
besides ,I want to investigate several results in coarse map, coarse equivalent
and coarse embedding. Further I have to proof some results of product of coarse
structure.
Coarse map need not be a continuous map. Coarse space has some applica tion in various parts in mathematics. More over coarse structure is a large scale
property so we can invest some results related with coarse space and topology.
Topology is the small scale structure, but topological coarse structure is the
large scale structure. We investigated some results about coarse maps, coarse
equivalent and coarse embedding.
