The open sets of a topological space together with union and intersection form a special kind of partially ordered set — a complete lattice in which finite meets distribute over arbitrary joins. These algebraic gadgets are called locales and as it turns out they behave very similar to the classical spaces we know from point set topology — with one major exception: There are interesting locales which have no points at all!
We will give a brief glimpse into this pointfree world and hopefully convince you that giving up points is not as bad an idea as one might think.