Consider the Jordan curve theorem. A simple closed curve (that is, the topological image of a circle) lying in the plane divides the plane into precisely two regions and forms their common boundary.
The beginner/reader/student might say but so what …this is too obvious…is topology used to prove such silly obvious results only?
Well, be that as it may, even these “obvious” things are to be taken much more seriously: one can easily give examples of propositions which sound as “obvious” as the Jordan curve theorem, but which may be proved false. Who would believe, for example, that in a plane there are three (four, five, …in fact, infinitely many) simply connected bounded regions which all have the same boundary or that one can find in three dimensional space a simple Jordan arc (that is, a topological image of a polygonal line) such that there are circles outside of this arc that cannot possibly be contracted to a point without meeting it? There are also closed surfaces of genus zero which possess an analogous property. In other words, one can construct a topological image of a sphere and an ordinary circle in its interior in such a way that the circle may not be contracted to a point wholly inside the surface?
Cheers, cheers, cheers,