How is general topology different from algebraic topology?

General topology develops the point-set foundations — open sets, continuity, compactness, connectedness — while algebraic topology assigns algebraic invariants such as homotopy and homology groups to spaces to distinguish them up to deformation.

Why define topology with open sets instead of distance?

Many important spaces (quotients, function spaces, abstract product spaces) carry no natural metric, yet still have a well-defined notion of continuity; the open-set axioms capture continuity in this fully general setting.

General Topology

General topology studies spaces defined by a notion of nearness — open sets — and the continuous maps between them, providing the foundational language of limits, convergence, and continuity for the rest of geometry and analysis.

Definition

A topology on a set X is a collection of subsets (the open sets) containing the empty set and X and closed under arbitrary unions and finite intersections; general topology is the study of such spaces and the continuous functions between them.

Scope

This area covers the abstract framework of topological spaces: how a topology is specified (open sets, bases, subbases), how continuity and homeomorphism are defined without reference to distance, and the global properties that distinguish spaces, chiefly compactness, connectedness, and the separation hierarchy. It includes product, subspace, and quotient constructions and metrization results that connect abstract topologies back to metric spaces. It excludes the algebraic invariants of algebraic topology and the smooth structure of differential geometry, which build on this foundation.

Sub-topics

Core questions

What minimal data specifies a notion of continuity on a set, independent of any metric?
Which topological properties are preserved under continuous maps, products, subspaces, and quotients?
When can an abstract topological space be realized as a metric space (metrization)?
How do compactness and connectedness encode the global shape and finiteness behavior of a space?

Key concepts

Open and closed sets, neighborhoods, interior and closure
Basis and subbasis for a topology
Continuous maps, homeomorphisms, and topological invariants
Subspace, product, and quotient topologies
Compactness, connectedness, and the separation axioms

Clinical relevance

General topology is the common substrate of modern mathematics: it supplies the rigorous meaning of convergence and continuity used in analysis, the spaces underlying functional analysis and differential geometry, and the point-set prerequisites assumed throughout algebraic topology.

History

Point-set topology grew from late-19th and early-20th-century efforts to abstract the notion of continuity from the real line, crystallizing in Hausdorff's 1914 axiomatization of topological spaces and maturing into the standardized curriculum codified by mid-century texts such as Kelley (1955) and Munkres.

Key figures

Felix Hausdorff
James Munkres
John L. Kelley

Seminal works

munkres2000
kelley1955

Frequently asked questions

How is general topology different from algebraic topology?: General topology develops the point-set foundations — open sets, continuity, compactness, connectedness — while algebraic topology assigns algebraic invariants such as homotopy and homology groups to spaces to distinguish them up to deformation.
Why define topology with open sets instead of distance?: Many important spaces (quotients, function spaces, abstract product spaces) carry no natural metric, yet still have a well-defined notion of continuity; the open-set axioms capture continuity in this fully general setting.