metric spaces and the topological groups, every uniform space is completely regular. The converse is also true: every completely regular space is uniformisable. Every CW complex is Tychonoff. Every normal regular space is completely regular, and every normal Hausdorff space is Tychonoff. The Niemytzki plane is an example of a Tychonoff space that is not normal. Properties Preservation Complete regularity and the Tychonoff property are well-behaved with respect to initial topologies. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property is preserved by taking point-separating initial topologies. It follows that: Every subspace of a completely regular or Tychonoff space has the same property. A nonempty product space is completely regular (respectively Tychonoff) if and only if each factor space is completely regular (respectively Tychonoff). Like all separation axioms, complete regularity is not preserved by taking final topologies. In particular, quotients of completely regular spaces need not be regular. Quotients of Tychonoff spaces need not even be Hausdorff, with one elementary counterexample being the bug-eyed line. There are closed quotients of the Moore plane that provide counterexamples. Real-valued continuous functions For any topological space let denote the family of real-valued continuous functions on and let be the subset of bounded real-valued continuous functions. Completely regular spaces can be characterized by the fact that their topology is completely determined by or In particular: A space is completely regular if and only if it has the initial topology induced by or A space is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in (i.e. the zero sets form a basis for the closed sets of ). A space is completely regular if and only if the cozero sets of form a basis for the topology of Given an arbitrary topological space there is a universal way of associating a completely regular space with Let ρ be the initial topology on induced by or, equivalently, the topology generated by the basis of cozero sets in Then ρ will be the finest completely regular topology on that is coarser than This construction is universal in the sense that any continuous function to a completely regular space will be continuous on In the language of category theory, the functor that sends to is left adjoint to the inclusion functor CReg → Top. Thus the category of completely regular