Home > Open and closed maps

Note that neither open nor closed maps are required to be continuous. Although their definitions seem natural, open and closed maps are much less important than continuous maps. Recall that a function f : X → Y is continuous if the preimage of any open set of Y is open in X, or equivalently: if the preimage of every closed set of Y is closed in X.

1 Examples

Every homeomorphism is open, closed, and continuous. In fact, a bijective continuous map is a homeomorphism iff it's open, or equivalently, iff it's 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).

Whenever we have a product of topological spaces X=ΠX_i, then 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. (Note that product projections need not be closed. 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.)

To every point on the unit circleIllustration of a unit circle. t is an angle measure. In mathematics, a unit circle is a circle with unit radius, i. a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, we can associate the angleThis article is about angles in geometry. For other articles, see Angle (disambiguation An angle (from the Lat. angulus a corner, a diminutive, of which the primitive form, angus does not occur in Latin; cognate are the Lat. angere, to compress into a ben of the positive x-axis with the ray connecting the point with the origin. This function from the unit circle to the half-open intervalTopology In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. Interval notation is where the permitted values for a variable are expressed as ranging over an in [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact spaceIn mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space R n in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if e 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 codomainSet theory Given a function , the set B is called the codomain of f. The codomain is not to be confused with the range f(A), which is in general only a subset of B. Example Let the function f be a function on the real numbers: : defined by : The codomain is essential!

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

The floor functionIn mathematics, the floor function is the function defined as follows: for a real number x floor x is the largest integer less than or equal to x''. For example, floor(2. 9) 2, floor(-2) -2 and floor(-2. The floor function is also denoted by x or. A more from RIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may to ZThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st is open and closed (because Z carries the discrete topology). This example shows that the image of a connected space under an open or closed map need not be connected.