nLab
continuous map

Continuous maps

Idea

A continuous map is a morphism in the category of topological spaces and in some similar contexts like locales and convergence spaces. (See also continuous space.)

Definitions

A function f:XYf\colon X\to Y between topological spaces is a continuous map (or is said to be continuous) if for every open subset UYU \subset Y, the preimage f 1(U)f^{-1}(U) is open in XX.

A function ff between convergence spaces is continuous if for any filter FF such that FxF \to x, it follows that f(F)f(x)f(F) \to f(x), where f(F)f(F) is the filter generated by the filterbase {F(A)|AF}\{F(A) \;|\; A \in F\}.

A continuous map between locales is simply a frame homomorphism in the opposite direction. Equivalently (via the adjoint functor theorem), it may be defined as a homomorphism of inflattices whose left adjoint preserves finitary meets (and hence is a frame homomorphism).

Properties

Since continuity is defined in terms of preservation of property (namely preserving “openness” under preimages), it is natural to ask what other properties they preserve.
Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.

Properties preserved

  1. By definition, the preimage of an open set is open.
  2. Similarly, the preimage of an closed set is closed.
  3. The image of a connected subset is again connected.
  4. The image of a compact subset is again compact.

Special maps

  1. The preimage of a compact set need not be compact; a continuous map for which this is true is known as a proper map.

  2. The image of an open set need not be open; a continuous map for which this is true is said to be an open map. (Technically, an open map is any function with just this property.)

  3. The image of an closed set need not be closed; a continuous map for which this is true is said to be an closed map. (Technically, a closed map is any function with just this property.)

  4. A continuous map of topological spaces which is invertible as a function of sets is a homeomorphism if the inverse function is a continuous map as well.

Special cases in specific contexts

Although these don’t make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as metric spaces:

Revised on August 25, 2014 04:57:01 by Toby Bartels (98.19.44.147)