Could someone explain to me this example? I know all the basis and definitions but I am new to calculation of cohomology group

here is the example

http://i51.tinypic.com/wb7609.jpg

Why ?? why do we get a real numbers and then we get just 0?

and then boundary group I guess that it is because a point does not have a boundary, is it correct and finally how did they get the cohomology group? how did they obtain again R and 0?

Thank you