General topology

General topology studies topological spaces and properties preserved by continuous maps, without requiring distance or smoothness. It provides a framework for continuity, convergence, and connectedness in abstract spaces.

Topological space: A set equipped with a collection of subsets, called open sets, satisfying axioms that define its topology. This is the basic object whose properties general topology studies.
Convergence: A relation describing when a sequence or more general indexed family approaches a limit. Topologies define convergence even when no distance function is available.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, identity of indiscernibles, and the triangle inequality. Every metric induces a topology, but general topology also studies spaces with no compatible metric.
Henri Poincaré: A French mathematician whose work helped establish topology as a distinct field around the turn of the twentieth century. His foundational work connected spatial questions to the emerging language of topology.
Fixed-point theorem: A theorem guaranteeing that a function maps at least one point to itself under specified conditions. Topological hypotheses can guarantee fixed points without requiring explicit solutions.
Open set: A member of a topology, with unions and finite intersections of such members remaining in the topology. Open sets encode the structure used to define continuity without distances.
Compact space: A topological space in which every open cover has a finite subcover. Compactness turns global covering conditions into finite ones.
Differential topology: The study of smooth manifolds and smooth maps, including their topological properties. It adds smoothness to spaces whose underlying topology can be studied independently.
Felix Hausdorff: A German mathematician who gave an influential axiomatic formulation of topological spaces in 1914. His axioms shaped the modern abstract definition of a topological space.
Brouwer fixed-point theorem: The theorem that every continuous map from a closed ball in finite-dimensional Euclidean space to itself has a fixed point. It is a landmark consequence of topological structure and continuity.

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