Thread: Simple Topology Equivalence Question

Originally Posted by Mad Cow

1) Are a rectangle and a rectangle with a hole equalivant?
2) Are an octagon and a rectangle equivilant?
3) Are a circle and a rectangle rectangle equivilant?

1) No
2) Yes
3) Yes

Originally Posted by Perspective

well, that all depends on how you define "equivalent". Is there more information to these questions? it all seems a little vague

Two sets A and B are equivalent in this sense if there exists a bijective continuous function from A to B.

Originally Posted by Perspective

but i guess the real question is why

The proof that a circle and a rectangle are topologically equivalent can be done by observing that
x = r cos t , 0<r<1
y = r sin t , 0<t<2pi
is a circle and the (r,t) space is a rectangle.