Lexicographic order topology on the unit square
As the name suggests, we use the lexicographical ordering on the square to define a topology. Given two points in the square, say (x,y) and (u,v), we say that (x,y) (u,v) if and only if either x < u or both x = u and y < v. Given the lexicographical ordering on the square, we use the order topology to define the topology on S.
The order topology makes S into a completely normal Hausdorff space. It is an example of an order topology in which there are uncountably many pairwise-disjoint homeomorphic copies of the real line. Since the lexicographical order on S can be proven to be complete, then this topology makes S into a compact set. At the same time, S is not separable, since the set of all points of the form (x,1/2) is discrete but is uncountable. Hence S is not metrizable (since any compact metric space is separable); however, it is first countable. Also, S is connected but not path connected, nor is it locally path connected. Its fundamental group is trivial.