Aureolux said:It can't be saying something like that, considering G is a subset of the product space (how can that be equal to a function?). All it means is any element of G [which, as a subset of the product space, has the form (x,y)] satisfies y=f(x), and any element in X x Y not in G does not satisfy this.
Although there are a few ways to show this, I will get you started on the way I think is most fundamental: assume there is an (x,y) in X x Y not in G. As I stated above this means y =/= f(x). Using the fact that Y is Hausdorff, you can form disjoint neighborhoods U and V around y and f(x), respectively. Try to use this to show that (X x Y) - G is open, and thus that G is closed. (Hint: we haven't used the continuity of f yet)
Damascus Road said:Let [tex] f: X \rightarrow Y [/tex] be a function. The graph of f is a subset of X x Y given by [tex] G = {(x,f(x) | x \in X } [/tex]. Show that if f is continuous and Y is Hausdorff, then G is closed in X x Y.
Any tips on how to start?
Is it saying that [tex] f: X \rightarrow Y = G [/tex] ?
Being Hausdorff means that for any two distinct points in the space, there exist disjoint open sets containing each of the points. This allows for a clear separation between points in the space.
To prove that a space is Hausdorff, one must show that for any two distinct points in the space, there exist disjoint open sets containing each of the points. This can be done by constructing such open sets or by using the definition of Hausdorff and logical reasoning.
Continuity is crucial in proving that G is closed, as it ensures that the inverse image of a closed set in the output space is a closed set in the input space. This is necessary for proving that G, the graph of a continuous function, is closed in the Cartesian product of the input and output spaces.
To prove that a function is continuous, one must show that the limit of the function as the input approaches a given point is equal to the value of the function at that point. This can be done using the epsilon-delta definition of continuity or by using other equivalent definitions and theorems.
Yes, a space can be Hausdorff without being continuous. Hausdorffness only concerns the separation of points in the space, while continuity refers to the behavior of a function on that space. Therefore, it is possible for a space to exhibit one property without the other.