Could someone explain topologically equivalent to me. I am to figure out if a pretzel is topologically equivalent to a pencil, a pipe or a trophy and I just don't understand please help

Answer 1

To explain topogically equivalent in the easiest terms is to suppose one object is made up of very soft and flexible material, or rubber, or something even more malleable, like playdough.
If you can change the shape of the object to make it become another, by deforming, twisting, or stretching, but without tearing or adding a branch, or a joint, then the two objects are topologically equivalent.

For example, a box and a pencil are topogically equivalent. A doughnut and a pencil are not.

Another famous example is the problem of the 7 bridges of Konigsberg, which is topologically equivalent to a network of nodes and edges. For more details, see:
http://www.mhhe.com/math/ltbmath/bennett_nelson/conceptual/netgraphs/graphs.htm