TITLE: Topology of a circle
QUESTION [6 upvotes]: I'm currently trying to learn topology by myself, before I started I knew topology had something to do with shaped stretched but not torn or glued. Now that I have started to learn I see talk about open and closed sets, neighborhoods, limits and such, but I don't understand how this relates to my previous understanding of topology.
I would be interested to know what is the topology, or the topological space of a circle (or the equivalent of a circle in topology, as a circle is the same as a square and many other things).
Thanks.

REPLY [2 votes]: At the heart of it all is the "topological idea of a circle", or of a loop for short. A loop does not have to look like the circle $x^2+y^2=1$ in the plane, but even a four year old child recognizes a loop when he sees one, and complains when it is fenced in by such a loop. In order to define precisely what a loop is we have to set up a standard model of a loop, as well as  a procedure to decide whether any particular geometric object presented to us qualifies as a loop. As standard model we could take the unit circle $S^1$ in the $(x,y)$-plane mentioned above, or the quotient set ${\mathbb R}/{\mathbb Z}$ of real numbers modulo $1$. I think you could accept both of these as loops in an intuitive way.
Now it comes to a formalization of "shaped stretched but not torn or glued", necessary for a mathematically precise procedure to test the "loopness" of a given geometric object. That's where the notions of general topology: open, closed, continuous, homeomorphism, etc., come in. Note that mathematics needed some 2500 years to come up with a definitive version of these notions, even though already Euclid knew what a loop is.