Continuous function (topology)

Table of Contents

• 1 Definitions
  • 1.1. Open and closed set definition
  • 1.2. Neighborhood definition
  • 1.3. Closure and interior operator definition
  • 1.4. Closeness relation definition
• 2 Useful properties of continuous maps
• 3 Other notes

If a topological space has the metric topology , the two definitions coincide.

Given a set X a partial ordering can be defined on the possible topologies on X . A continuous functions between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space . Thus we can consider the continuity of a given function a topological property , depending only on the topologies of its domain and codomain spaces. A continuous function can be visualized as weakening the topology of the domain space.

In real analysis continuity of functions is commonly defined using the ε-δ definition which builds on the property of the real line being a metric space . As topological spaces generally do not have this property a more general definition is needed which reduces to the ε-δ definition in case of the real line.

The most common one defines continuous functions as those functions where the preimage s of open set s are open . Similar to the open set formulation is the closed set formulation , which says that preimage s of closed set s 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 x \in X if for any neighborhood V of f ( x ), there is a neighborhood U of x such that f(U) \subseteq V . 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 x \in X , then we simply say f is continuous.

In a metric space , it is equivalent to consider the neighbourhood system of open ball s centered at x and f ( x ) instead of all neighborhoods. This leads to the standard ε-δ 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.

• 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 first-countable space , then the converse also holds: any function taking limits of sequences to limits of sequences is continuous. In particular, the converse holds if X is a metric space . When using nets instead of sequences, this converse holds for a general topological space X .

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 T 0 , 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.

A continuous bijection is a homeomorphism if its domain is compact and its codomain is Hausdorff .