Previously we talked about bijections as a way of pairing up all the elements of two sets. Often this is done to express that one set is equivalent to another.
Once you have structure on the set, it isn't enough to just have a bijection. The elements of the two sets must be paired up in a way that maintains the structure before the two structured sets can be said to be equivalent.
Two topological spaces are equivalent if the bijection maintains the open sets. In other words, if the bijection maps open sets to open sets then our two spaces are topologically equivalent.
Another word for topologically equivalent is homeomorphic (note the 'e') and the topology-preserving mapping is called a homeomorphism.
A topological space is the most general space that has a notion of continuity, so two spaces that differ in terms of other structures (like distance between their points) might still be homeomorphic if continuity is preserved. One way to think about this is moulding a ball of clay...
Imagine taking a ball of clay and squashing it flat. If you think of the clay as a metric space, you've clearly changed the space quite a bit because distances between pairs of points are no longer the same. However, you haven't changed the topology. The open sets are still open sets in the squashed version. Squashing the clay is a homeomorphism. If you'd drawn a continuous line on your ball it would still be continuous after the squashing. Squashing hasn't ripped two points apart from one another.
But, now consider pushing your thumb through the clay to mould it into a doughnut-shape. To make the hole, you had to rip points apart from one another. This has altered the open sets. Two points that might have been very close (and hence in some very small open sets together) might now only share very large open sets in common. Because the topology is not preserved, the mapping from ball to doughnut is not a homeomorphism.
A topologist would say that the ball of clay is not homeomorphic to the doughnut-shaped clay.
We've reached an important milestone because the Poincare Conjecture has to do with whether one particular type of topological space is always homeomorphic to another particular type.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.