Pistejoukkooppi

Pistejoukkooppi, or point-set topology, is a branch of topology that studies the properties of topological spaces that depend only on the arrangement of open sets and related structures, rather than on any particular metric or geometric form.

A topological space consists of a set X equipped with a family τ of subsets of X, called the open sets, satisfying: the empty set and X are in τ; the union of any collection of sets in τ is in τ; and the intersection of any finite number of sets in τ is in τ. The pair (X, τ) defines the topology of the space.

Common examples include the real numbers with the standard topology, the discrete topology where every subset is open, and the indiscrete topology where only the empty set and X are open. Any metric space induces a topology via open balls, linking point-set topology to analysis.

Key notions include continuity (maps that pull back open sets to open sets), bases and subspace topology, the product topology on product spaces, and quotient topology for identifying points. Convergence is described by sequences in metric spaces and more generally by nets or filters in arbitrary spaces.

Pistejoukkooppi also studies properties preserved by homeomorphisms, such as separation axioms (T0, T1, T2), countability axioms (first and second countable), compactness via open covers, and connectedness. In metric spaces, many questions reduce to sequences, while nets and filters handle convergence in general spaces.

The field developed in the late 19th and early 20th centuries with contributions from Fréchet, Hausdorff, and others, forming a foundational framework for analysis, geometry, and later areas such as functional analysis and algebraic topology.