Continuous function (topology)

In topology, a continuous function is generally defined as one for which preimages of open sets are open. Continuous functions are fundamental in describing the relationships between topological spaces, and allow simple generalizations of many results from real analysis to be proven. Because this definition only "uses" open sets, this makes continuity of a function a topological property, depending only on the topologies of its domain and range spaces.

Formulations of Continuity

Several equivalent formulations of continuity can be made, and each is useful in different situations. Similar to the open set formulation is the closed set formulation, which says that preimages of closed sets are closed.

Definition based on preimages are often difficult to use directly. Instead, suppose we have a function f from X to Y, where X,Y are topological spaces. We say f is continuous at x for some if for any neighborhood V of f(x), there is a neighborhood U of x such that . Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can find a small U containing x that will map inside it. If f is continuous at every , then we simply say f is continuous.

In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. This leads to the standard delta-epsilon definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance.

Useful properties of continuous maps

Some facts about continuous maps between topological spaces:

• If f : X → Y and g : Y → Z are continuous, then so is the composition g o f : X → Z.
• If f : X → Y is continuous and
  • X is compact, then f(X) is compact.
  • X is connected, then f(X) is connected.
  • X is path-connected, then f(X) is path-connected.

• If f : X → Y is continuous and a sequence (x_n) in X converges to a limit x, then the sequence (f(x_n)) obtained by applying f to each element converges to f(x). We say continuous functions take limits to limits. This also holds if sequences are replaced by general nets.
  • If X is a metric space, then the converse also holds: any function taking limits to limits is continuous. When using nets instead of sequences, this converse holds for a general topological space X.

Other notes

If a set is given the discrete topology, all functions with that space as a domain are continuous. If the domain set is given the indiscrete topology and the range set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open.

If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection need not be continuous, but if it is, this special function is called a homeomorphism.