Open and Closed Maps - Examples

Examples

Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism if and only if it is open, or equivalently, if and only if it is closed.

If Y has the discrete topology (i.e. all subsets are open and closed) then every function f : X → Y is both open and closed (but not necessarily continuous). For example, the floor function from R to Z is open and closed, but not continuous. This example shows that the image of a connected space under an open or closed map need not be connected.

Whenever we have a product of topological spaces X=ΠX_i, the natural projections p_i : X → X_i are open (as well as continuous). Since the projections of fiber bundles and covering maps are locally natural projections of products, these are also open maps. Projections need not be closed however. Consider for instance the projection p₁ : R2 → R on the first component; A = {(x,1/x) : x≠0} is closed in R2, but p₁(A) = R − {0} is not closed. However, for compact Y, the projection X × Y → X is closed. This is essentially the tube lemma.

To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying the codomain is essential.

The function f : R → R with f(x) = x2 is continuous and closed, but not open.