Hausdorff Space

Hausdorff space, in mathematics, type of topological space named for the German mathematician Felix Hausdorff. A topological space is a generalization of the notion of an object in three-dimensional space. It consists of an abstract set of points along with a specified collection of subsets, called open sets, that satisfy three axioms: (1) the set itself and the empty set are open sets, (2) the intersection of a finite number of open sets is open, and (3) the union of any collection of open sets is an open set. A Hausdorff space is a topological space with a separation property: any two distinct points can be separated by disjoint open sets—that is, whenever p and q are distinct points of a set X, there exist disjoint open sets U_p and U_q such that U_p contains p and U_q contains q.

The real number line becomes a topological space when a set U of real numbers is declared to be open if and only if for each point p of U there is an open interval centred at p and of positive (possibly very small) radius completely contained in U. Thus, the real line also becomes a Hausdorff space since two distinct points p and q, separated a positive distance r, lie in the disjoint open intervals of radius r/2 centred at p and q, respectively. A similar argument confirms that any metric space, in which open sets are induced by a distance function, is a Hausdorff space. However, there are many examples of non-Hausdorff topological spaces, the simplest of which is the trivial topological space consisting of a set X with at least two points and just X and the empty set as the open sets. Hausdorff spaces satisfy many properties not satisfied generally by topological spaces. For example, if two continuous functions f and g map the real line into a Hausdorff space and f(x) = g(x) for each rational number x, then f(x) = g(x) for each real number x.

Hausdorff included the separation property in his axiomatic description of general spaces in Grundzüge der Mengenlehre (1914; “Elements of Set Theory”). Although later it was not accepted as a basic axiom for topological spaces, the Hausdorff property is often assumed in certain areas of topological research. It is one of a long list of properties that have become known as “separation axioms” for topological spaces.