topologiques

Topologiques is a term rooted in mathematics, referring to topology and its related concepts. In its broad sense, topology studies properties of spaces that are preserved under continuous deformations, such as stretching or bending, but not tearing or gluing.

The central notion is the topological space. A topological space consists of a set X together with a topology τ, which is a collection of subsets of X called open sets that satisfy three axioms: the empty set and X are open; arbitrary unions of open sets are open; and finite intersections of open sets are open. The complements of open sets are called closed. Open and closed sets provide the language to describe continuity, convergence, and limits in a way that is independent of distances.

A function between topological spaces is continuous if the preimage of every open set is open. When a continuous bijection has a continuous inverse, it is a homeomorphism, and the two spaces are considered topologically the same. This equivalence relation captures the idea of properties that do not change under deformation.

Core examples include the standard topology on the real numbers, where open sets are unions of open intervals; the discrete topology, where every subset is open; and the indiscrete topology, where only the empty set and the whole space are open. More advanced constructions include subspace topologies, which inherit a topology from a larger space, and product topologies, which describe the topology on Cartesian products.

Key concepts and invariants include connectedness, compactness, and separation axioms (such as T0, T1, T2). Topologiques also intersects with areas like analysis, geometry, dynamical systems, and data analysis, particularly in the study of shapes, continuity, and qualitative properties of spaces.