homeomorphism. If every fiber of is a discrete subspace of then this open set is necessarily a subset of In particular, if then a conclusion that may be false without the assumption that 's fibers are discrete (see this footnote for an example). One corollary is that every continuous open surjection between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that is a dense open subset of its domain). For example, the map defined by the polynomial is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset is dense in with additional effort (using the inverse function theorem for instance), it can be shown that which confirms that this set is indeed dense in This example also shows that it is possible for to be a dense subset of 's domain. Because every fiber of every non-constant polynomial is finite (and thus a discrete, and even compact, subspace), this example generalizes to such polynomials whenever the mapping induced by it is an open map. Local homeomorphisms and Hausdorffness There exist local homeomorphisms where is a Hausdorff space but is not. Consider for instance the quotient space where the equivalence relation on the disjoint union of two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy. The two copies of are not identified and they do not have any disjoint neighborhoods, so is not Hausdorff. One readily checks that the natural map is a local homeomorphism. The fiber has two elements if and one element if Similarly, it is possible to construct a local homeomorphisms where is Hausdorff and is not: pick the natural map from to with the same equivalence relation as above. Properties A map is a local homeomorphism if and only if it is continuous, open, and locally injective. In particular, every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism. Whether or not a function is a local homeomorphism depends on its codomain. The image of a local homeomorphism is necessarily an open subset of its codomain and will also be a local homeomorphism (that is, will continue to be a local homeomorphism when it is considered as the surjective map onto its image, where has the subspace topology inherited from