 in mathematics, general 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. another name for general topology is point-set 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 words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. 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 distances can be assigned a number called a metric. having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces. == history == 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. ==a topology on a set==  let x be a set and let Ï be a family of subsets of x. then Ï is called a topology on x if:munkres, james r. topology. vol. 2. upper saddle river: prentice hall, 2000.adams, colin conrad, and robert david franzosa. introduction to topology: pure and applied. pearson prentice hall, 2008. # both the empty set and x are elements of Ï # any union of elements of Ï is an element of Ï # any intersection of finitely many elements of Ï is an element of Ï if Ï is a topology on x, then the pair (x, Ï) is called a topological space. the notation xÏ may be used to denote a set x endowed with the particular topology Ï. the 
General_topology