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General topology

In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.

The fundamental concepts in point-set topology are continuity, compactness, and connectedness:


The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.


Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.

the detailed study of subsets of the (once known as the topology of point sets; this usage is now obsolete)

real line

the introduction of the concept

manifold

the study of , especially normed linear spaces, in the early days of functional analysis.

metric spaces

General topology grew out of a number of areas, most importantly the following:


General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

There exist numerous topologies on any given . Such spaces are called finite topological spaces. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.

finite set

Every has a natural topology, since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn.

manifold

The is defined algebraically on the spectrum of a ring or an algebraic variety. On Rn or Cn, the closed sets of the Zariski topology are the solution sets of systems of polynomial equations.

Zariski topology

A has a natural topology that generalises many of the geometric aspects of graphs with vertices and edges.

linear graph

Many sets of in functional analysis are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.

linear operators

Any has a topology native to it, and this can be extended to vector spaces over that field.

local field

The is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics.

Sierpiński space

If Γ is an , then the set Γ = [0, Γ) may be endowed with the order topology generated by the intervals (a, b), [0, b) and (a, Γ) where a and b are elements of Γ.

ordinal number

X is , then f(X) is compact.

compact

X is , then f(X) is connected.

connected

X is , then f(X) is path-connected.

path-connected

X is , then f(X) is Lindelöf.

Lindelöf

X is , then f(X) is separable.

separable

X is , or Kolmogorov, if any two distinct points in X are topologically distinguishable. (It is a common theme among the separation axioms to have one version of an axiom that requires T0 and one version that doesn't.)

T0

X is , or accessible or Fréchet, if any two distinct points in X are separated. Thus, X is T1 if and only if it is both T0 and R0. (Though you may say such things as T1 space, Fréchet topology, and Suppose that the topological space X is Fréchet, avoid saying Fréchet space in this context, since there is another entirely different notion of Fréchet space in functional analysis.)

T1

X is , or T2 or separated, if any two distinct points in X are separated by neighbourhoods. Thus, X is Hausdorff if and only if it is both T0 and R1. A Hausdorff space must also be T1.

Hausdorff

X is , or Urysohn, if any two distinct points in X are separated by closed neighbourhoods. A T2½ space must also be Hausdorff.

T2½

X is , or T3, if it is T0 and if given any point x and closed set F in X such that x does not belong to F, they are separated by neighbourhoods. (In fact, in a regular space, any such x and F is also separated by closed neighbourhoods.)

regular

X is , or T3½, completely T3, or completely regular, if it is T0 and if f, given any point x and closed set F in X such that x does not belong to F, they are separated by a continuous function.

Tychonoff

X is , or T4, if it is Hausdorff and if any two disjoint closed subsets of X are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is Urysohn's lemma.)

normal

X is , or T5 or completely T4, if it is T1 and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.

completely normal

X is , or T6 or perfectly T4, if it is T1 and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff.

perfectly normal

Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous.


Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles.


In all of the following definitions, X is again a topological space.


The Tietze extension theorem: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space.

: a set is open if every sequence convergent to a point in the set is eventually in the set

sequential space

: every point has a countable neighbourhood basis (local base)

first-countable space

: the topology has a countable base

second-countable space

: there exists a countable dense subspace

separable space

: every open cover has a countable subcover

Lindelöf space

: there exists a countable cover by compact spaces

σ-compact space

An axiom of countability is a property of certain mathematical objects (usually in a category) that requires the existence of a countable set with certain properties, while without it such sets might not exist.


Important countability axioms for topological spaces:


Relations:

List of examples in general topology

for detailed definitions

Glossary of general topology

for related articles

List of general topology topics

Category of topological spaces

Topologie Générale (General Topology), ISBN 0-387-19374-X.

Bourbaki

(1955) General Topology, link from Internet Archive, originally published by David Van Nostrand Company.

John L. Kelley

General Topology, ISBN 0-486-43479-6.

Stephen Willard

Topology, ISBN 0-13-181629-2.

James Munkres

Introduction to Topology and Modern Analysis, ISBN 1-575-24238-9.

George F. Simmons

Topology: Point-Set and Geometric, ISBN 0-470-09605-5.

Paul L. Shick

General Topology, ISBN 3-88538-006-4.

Ryszard Engelking

; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446

Steen, Lynn Arthur

O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, , ISBN 978-0-8218-4506-6.

Elementary Topology: Textbook in Problems

Some standard books on general topology include:


The arXiv subject code is math.GN.

Media related to General topology at Wikimedia Commons