spaces and is if the preimage of every open set in is open in The function is called if the image of every open set in is open in An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals. Notes and cautions "Open" is defined relative to a particular topology Whether a set is open depends on the topology under consideration. Having opted for greater brevity over greater clarity, we refer to a set X endowed with a topology as "the topological space X" rather than "the topological space ", despite the fact that all the topological data is contained in If there are two topologies on the same set, a set U that is open in the first topology might fail to be open in the second topology. For example, if X is any topological space and Y is any subset of X, the set Y can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the intersection of Y with an open set from the original topology on X." This potentially introduces new open sets: if V is open in the original topology on X, but isn't open in the original topology on X, then is open in the subspace topology on Y. As a concrete example of this, if U is defined as the set of rational numbers in the interval then U is an open subset of the rational numbers, but not of the real numbers. This is because when the surrounding space is the rational numbers, for every point x in U, there exists a positive number a such that all points within distance a of x are also in U. On the other hand, when the surrounding space is the reals, then for every point x in U there is positive a such that all points within distance a of x are in U (because U contains no non-rational numbers). Generalizations of open sets Throughout, will be a topological space. A subset of a topological space is called: if , and the complement of such a set is called . , , or if it satisfies any of the following equivalent conditions: There exists subsets such that is open in is a dense subset