In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
The fundamental concepts in point-set topology are continuity, compactness, and connectedness:
Continuous functions, intuitively, take nearby points to nearby points.
Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
Connected sets are sets that cannot be divided into two pieces that are far apart.
The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.
Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.
General topology grew out of a number of areas, most importantly the following:
the detailed study of subsets of the real line (once known as the topology of point sets; this usage is now obsolete)
the introduction of the manifold concept
the study of metric spaces, especially normed linear spaces, in the early days of functional analysis.
General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.
Topological space
Let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
Both the empty set and X are elements of τ
Any union of elements of τ is an element of τ
Any intersection of finitely many elements of τ is an element of τ
If τ is a topology on X, then the pair (X, τ) is called a topological space.
