Point-set topology rests on continuity, compact, connected
General topology, also called point-set topology, supplies the basic set-theoretic definitions that differential, geometric and algebraic topology all build on. Its three central ideas are continuity, compactness and connectedness, and all three turn out to depend on a single choice: which sets count as open.
Informally, a continuous function never tears close points apart, a compact set can be covered by a finite number of pieces however small you make them, and a connected set cannot be broken into two separated parts. Words like close, small and separated gain exact meaning through open sets. Change what counts as open and the continuous functions, compact sets and connected sets all change with it; each such choice is called a topology, and a set carrying one is a topological space.
The rules are brief. A family of subsets is a topology if it includes the empty set and the whole set, and is closed under arbitrary unions and finite intersections. A set is closed when its complement is open, and some sets are both, called clopen, while others are neither. A base is a collection of open sets whose unions give every open set, and subspace, product and quotient topologies build new spaces from old ones.
One set can wear many topologies. In the discrete topology every subset is open, and only sequences that eventually stop moving converge. The trivial topology admits just two open sets, nothing and everything, so every sequence converges to every point, proof that limits need not be unique; Hausdorff spaces restore uniqueness. The cofinite topology is the smallest T1 topology on an infinite set, and the lower limit topology on the real line, built from half-open intervals, is strictly finer than the usual one.
The field grew out of the study of subsets of the real line and of metric spaces in early functional analysis, reaching its modern form around 1940. Metric spaces, where a distance is defined between points, simplify many proofs, and on the Zariski topology the closed sets are solution sets of polynomial equations.
Source: General topology
