graph theorems for topological spaces

We wish to show the relationMathworldPlanetmath between continuous mapsMathworldPlanetmath and their graphs is closer that it may look. Recall, that if f:XY is a function between sets, then the set Γ(f)={(x,f(x))X×Y} is called the graph of f.

PropositionPlanetmathPlanetmath 1. If f:XY is a continuous map between topological spacesMathworldPlanetmath such that Y is HausdorffPlanetmathPlanetmath, then the graph Γ(f) is a closed subset of X×Y in product topology.

Proof. Indeed, we will show, that Z=(X×Y)\Γ(f) is open. Let (x,y)Z. Then f(x)y and thus (since Y is Hausdorff) there exist open subsetes V1,V2Y such that f(x)V1, yV2 and V1V2=. Since f is continuousMathworldPlanetmath, then U=f-1(V1) is open in X.

Note, that the condition V1V2= implies, that f(U)V2=. Therefore U×V2 is a subset of Z. On the other hand this subset is open (since it is a productPlanetmathPlanetmath of two open sets) in product topology and (x,y)U×V2. This shows, that every point in Z belongs to Z together with a small neighbourhood, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Unfortunetly, the converseMathworldPlanetmath of this theorem is not true as we will see later. Nevertheless we can achieve similar result, if we assume a bit more about spaces:

Proposition 2. Let f:XY be a function, where X,Y are Hausdorff spaces with Y compactPlanetmathPlanetmath. If Γ(f) is a closed subset of X×Y in product topology, then f is continuous.

Proof. Let FY be a closed set. We will show that f-1(F) is also closed. Consider projections


They are both continuous and thus πY-1(F) is closed in X×Y. Since Γ(f) is also closed, then


is closed in X×Y. It is well known, that since Y is compact, then πX is a closed map (this is easily seen to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the tube lemma). Furthermore it is easy to see, that πX(Z)=f-1(F) and the proof is complete.

Counterexample. Let denote the set of reals (with standard topology). Consider function f: given by f(x)=1/x and f(0)=0. It is obvious, that f is discontinuousMathworldPlanetmath at x=0, but also it can be easily checked, that Γ(f) is closed in 2. Note, that is not compact.

Title graph theorems for topological spaces
Canonical name GraphTheoremsForTopologicalSpaces
Date of creation 2013-03-22 19:15:09
Last modified on 2013-03-22 19:15:09
Owner joking (16130)
Last modified by joking (16130)
Numerical id 7
Author joking (16130)
Entry type Theorem
Classification msc 54C05
Classification msc 26A15