topology (thing)

A mathematical object, a collection of subsets of a given set, usually interpreted as the structure of a space.

Given a set X, a collection T of subsets of X can be called a "topology for  X" if and only if

• X is an element of T,
• the empty set is an element of T,
• For any two elements of T, their intersection is in T,
• For any collection of elements of T, their union is in T.

A set X plus a topology T is called a topological space. Elements of a topology are usually labeled as open sets in the topology. However, it doesn't have to be this way.

A set C is closed in the topology if and only if X - C ∈ T.  The Kuratowski closure theorems show that if we start from a collection of "closed" sets such that

• X is closed,
• the empty set is closed,
• For any collection of closed sets, their intersection is closed,
• For any two closed sets, their union is also closed,

 T is still a topology. Thus, we can start from "closed" sets instead of "open" sets.

Given a particular topology  T for a set X, an open set U (i.e.  U _^∈ T) containing a particular element x of  X is called an open set about x or an open neighborhood of x. Analagous terms for closed sets apply.

Given a subset M of X, the interior of M is the smallest open set containing M. Similarly, the exterior of M is the smallest closed set containing M. The frontier of M is Exterior(M)-Interior(m).

A base (or basis) for a topology T of a set  X is a collection B of subsets of X such that each element of T is the union of elements of B.

A subbase ("subbasis") for a topology T of a set  X is a collection S of subsets of  X where every element of T is the union of sets that are the intersections of finitely many elements of  S. Every cover of X is a subbase for some topology of X. Because of this, any collection S of sets a is subbase for some topology of ∪S.

Each set X has at least two topologies:

• its discrete topology where every subset of X is open (that is, the power set of X), and
• its indiscrete topology {0, X}, also known as the trivial topology.

Topologies of a set X can be partially ordered using the subset relation.  Given two topologies T, S,

• T is coarser than S  means T ⊆ S,
• T is finer than S means T ⊇ S.

A set's trivial topology is the order's minimal element, that is, the coarsest topology of all, and its discrete topology is the order's maximal element, that is, the finest topology of all.

Topology first arose as a way to explain the structure of the real numbers.  An important topology on the real numbers is the Euclidean topology, which has open intervals in the real number line (the origin of the terms "open" and "closed") as a base.

Important properties of topologies include

• separability properties such as Hausdorff-ness, Lindelöf-ness, dense-ness, regularity, and normality.
• compactness,
• normalness and completeness.

Discussions of topology usually lead to discussions of:

• metric spaces and the metric topologies they induce.
• homotopy theory, the classification of manifolds, and other algrebraic aspects of topology,
• graph theory.