Since manifolds are locally compact Hausdorff spaces, manifolds are necessarily Tychonoff spaces. And a Tychonoff space is a topological space that is both Hausdorff and completely regular. This is cut and paste from wikipedia.org. Further, X is a completely regular space if given any closed set F and any point x that does not belong to F, then there is a continuous function f from X to the real line R such that f(x) is 0 and, for every y in F, f(y) is 1. And so my question is: does this mean that in order for this property of completely regularity to hold for a space that one must be able to construct a set F with a point x in F and y outside F with a function f as described above? And this must hold no matter how close x and y are to each other? Is there any limit to the smallness of the set F? Does this mean that the sets, F, must exist in the topology, or can be constructed from the underlying elments whether in the topology or not? Thanks.