nLab
topological space

Topological spaces

Idea

The notion of topological space aims to axiomatize the idea of a space as a collection of points that hang together (“cohere”) in a continuous way.

The proverbial motivating example is the surface of a torus and that of a mug: these are isomorphic as topological spaces, because, abstractly, the surface of the mug can be deformed continuously to become the standard torus: the continuous cohesion among the collections of points of the two surfaces is the same.

There is a slight generalization of the notion of topological space to that of a locale, which consists of dropping the assumption that all neighbourhoods are explicitly or even necessarily supported by points. In this form the definition is quite fundamental and can be naturally motivated from just pure logic – as the formal dual of frames – as well as, and dually, from category theory in its variant as topos theory – by the notion of (0,1)-toposes.

Topological spaces are the objects studied in topology. By equipping them with a notion of weak equivalence, namely of weak homotopy equivalence, they turn out to support also homotopy theory.

Topological spaces equipped with extra property and structure form the fundament of much of geometry. For instance a topological space locally isomorphic to a Cartesian space is a manifold. A topological space equipped with a notion of smooth functions into it is a diffeological space. The intersection of these two notions is that of a smooth manifold on which differential geometry is based. And so on.

Definitions

We present first the

and then a list of different

Finally we mention genuine

Standard definition

Definition

A topological space is a set XX equipped with a set of subsets UXU \subset X, called the open sets, which are closed under

  1. finite intersections
  2. arbitrary unions.
Remark

The word ‘topology’ sometimes means the study of topological spaces but here it means the collection of open sets in a topological space. In particular, if someone says ‘Let TT be a topology on XX’, then they mean ‘Let XX be equipped with the structure of a topological space, and let TT be the collection of open sets in this space’.

Remark

Since XX itself is the intersection of zero subsets, it is open, and since the empty set \emptyset is the union of zero subsets, it is also open. Moreover, every open subset UU of XX contains the empty set and is contained in XX

UX, \emptyset \subset U \subset X \,,

so that the topology of XX is determined by a poset of open subsets Op(X)Op(X) with bottom element =\bot = \emptyset and top element =X\top = X.

Since by definition the elements in this poset are closed under finite meets (intersection) and arbitrary joins (unions), this poset of open subsets defining a topology is a frame, the frame of opens of XX.

Definition

A homomorphisms between topological spaces f:XYf : X \to Y is a continuous function:

a function f:XYf:X\to Y of the underlying sets such that the preimage of every open set of YY is an open set of XX.

Topological spaces with continuous maps between them form a category, usually denoted Top.

Remark

The definition of continuous function f:XYf : X \to Y is such that it induces a homomorphism of the corresponding frames of opens the other way around

Op(X)Op(Y):f 1. Op(X) \leftarrow Op(Y) : f^{-1} \,.

And this is not just a morphism of posets but even of frames. For more on this see at locale.

Alternate equivalent definitions

There are many equivalent ways to define a topological space. A non-exhaustive list follows:

Variations

The definition of topological space was a matter of some debate, especially about 100 years ago. Our definition is due to Bourbaki, so may be called Bourbaki spaces.

For some purposes, including homotopy theory, it is important to use nice topological spaces (such as sequential topological spaces) and/or a nice category of spaces (such as compactly generated spaces), or indeed to directly use a model of \infty-groupoids (such as simplicial sets).

On the other hand, when doing topos theory or working in constructive mathematics, it is often more appropriate to use locales than topological spaces.

Some applications to analysis require more general convergence spaces or other generalisations.

Examples

Special cases

Specific examples

Example

The Cartesian space n\mathbb{R}^n with its standard notion of open subsets generated from: unions of open balls D n nD^n \subset \mathbb{R}^n.

References

See at topology.

Revised on April 13, 2014 05:46:53 by Urs Schreiber (185.37.147.12)