Of course, for many topological spaces the similarities are remote, but aid in judgment and guide proofs. Also, any set can be given the trivial topology(also called the indiscrete topology), in which only the empty set and the whole space are open.

We defineE©Ð\ß Ñg int the is open and }\Eœ interior of E in \œÖSÀS S©E×

A given set may have many different topologies. Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. In Abstract Algebra, a field generalizes the concept of operations on the real number line. Every sequence and net in this topology converges to every point of the space. Recall from the Hausdorff Topological Spaces page that a topological space is said to be a Hausdorff space if for every distinct there exists open neighbourhoods such that , and .

Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T 2) is the most frequently used and discussed. If a set is given a different topology, it is viewed as a different topological space. Examples include a closed interval, a rectangle, or a finite set of points. This general definition allows concepts about quite different mathematical objects to be grasped intuitively by comparison with the real numbers.

The only convergent sequences or nets in this topology are those that are eventually constant.

This applies, for example, to the definitions of interior, closure, and frontier in pseudometric spaces, so these definitions can also be carried over verbatim to a topological space Ð\ß ÑÞg Definition 2.7 Suppose . The prototype Let X be any metric space and take to be the set of open sets as defined earlier. Likewise, the concept of a topological space is concerned with generalizing the structure of sets in Euclidean spaces. In topology and related branches of mathematics, a Hausdorff space, separated space or T 2 space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other.

Examples.

The open sets are the sets U ⊂ R such that every point in U lies in an open interval wholly contained in U; in symbols

The most basic example is the space R with the order topology. Some "extremal" examples Take any set X and let = {, X}. Hausdorff Topological Spaces Examples 3. This notion is defined for more general topological spaces than Euclidean space in various ways.

The properties verified earlier show that is a topology.

Any set can be given the discrete topology in which every subset is open. We also looked at two notable examples of Hausdorff spaces - the first being the set of real numbers with the usual topology of open intervals on , and the second being the discrete topology … …

(X, ) is called a topological space. This is a topological vector space because: The topology is closed under arbitrary unions and finite intersections.