Topology

Definition: Topology is defined with respect to a set X. A 'topology in X' is a set of subsets of X satisfying several criteria. Let t denote a topology in X. The sets in t are by definition 'open sets' with respect to t, and sets outside of t are not. t satisfies the following:

• X and the null set are in t.
• Finite or infinite unions of open sets (that is, elements of t) are also in t.
• Finite intersections of open sets are in t.

More than one topology in X may be possible for a given set X.

The complement of a set in t is said to be a 'closed set'.

Element of X may be called 'points'.

A 'neighborhood' of a point x is any open set containing x.

Let M be a subset of X. A point x in X is a 'contact point' of M if every neighborhood of x contains at least one point of M; and x would be a 'limit point' of M if every neighborhood of x contained infinitely many points of M. The set of all contact points of M is the 'closure' of M.

A 'topological space' is a pair of sets (X, t) satisfying the above.

All metric spaces are topological spaces. The sets one would call open in a metric space satisfy the criteria above; one could also label all subsets of X as open for purpose of listing the members of the topology and they would then satisfy the definition above.

(Econterms)