This page is about topology as a field of mathematics. For topology as a structure on a set, see topological space.
Parts of this page exists also in a German language version, see at Topologie.
I believe that we lack another analysis properly geometric or linear which expresses location directly as algebra expresses magnitude.
G. W. Leibniz [letter to Huygens 1679, according to Bredon 93, p. 430]
Idea
Topology is one of the basic fields of mathematics. The term is also used for a particular structure in a topological space; see topological structure for that.
The subject of topology deals with the expressions of continuity and boundary, and studying the geometric properties of [originally: metric] spaces and relations of subspaces, which do not change under continuous deformations, regardless to other [such as in their metric properties].
Topology as a structure enables one to model continuity and convergence locally. More recently, in metric spaces, topologists and geometric group theorists started looking at asymptotic properties at large, which are in some sense dual to the standard topological structure and are usually referred to as coarse topology.
There are many cousins of the concept of topological spaces, e.g. sites, locales, topoi, higher topoi, uniformity spaces and so on, which specialize or generalize some aspect or structure usually found in Top.
One of the tools of topology, homotopy theory, has long since crossed the boundaries of topology and applies to many other areas, thanks to many examples and motivations as well as of abstract categorical frameworks for homotopy like Quillen model categories, Brown’s categories of fibrant objects and so on.
Introduction
The following gives a quick introduction to some of the core concepts and tools of topology:
A detailed introduction is going to be at Introduction to Topology.
Continuity
The key idea of topology is to study spaces with “continuous maps” between them. The concept of continuity was made precise first in analysis, in terms of epsilontic analysis of open balls, recalled as def. below. Then it was realized that this has a more elegant formulation in terms of the more general concept of open sets, this is prop. below. Adopting the latter as the definition leads to the concept of topological spaces, def. below.
First recall the basic concepts from analysis:
[metric space]
A metric space is
such that for all x,y,z∈Xx,y,z \in X:
-
d[x,y]=0⇔x=yd[x,y] = 0 \;\;\Leftrightarrow\;\; x = y
-
[symmetry] d[x,y]=d[y,x]d[x,y] = d[y,x]
-
[triangle inequality] d[x,y]+d[y,z]≥d[x,z]d[x,y]+ d[y,z] \geq d[x,z].
Every normed vector space [V,|−|][V, {\vert - \vert}] becomes a metric space according to def. by setting
d[x,y]≔|x−y|. d[x,y] \coloneqq {\vert x-y \vert} \,.
[epsilontic definition of continuity]
For [X,d X][X,d_X] and [Y,d Y][Y,d_Y] two metric spaces [def. ], then a function
f:X⟶Y f \;\colon\; X \longrightarrow Y
is said to be continuous at a point x∈Xx \in X if for every ϵ>0\epsilon \gt 0 there exists δ>0\delta\gt 0 such that
d X[x,y]