Need to understand the concept graph on metric spaces.

Question

The definition of graph I have been given is the following:

$δ(F)={[(x,y)∈X*Y:y∈F(x)]}$

Where $F$ is a correspondence $F:X→Y$.

I would be grateful if someone could explain its meaning to me in words.

Answer 1

For a standard function like $f(x)=x^2$ from $\Bbb R$ to $\Bbb R$, the graph is the set of pairs $\Bbb R^2$ "tracing the function": $\Gamma(f)=\{(x,f(x)), x \in \Bbb R\}$, the probably familiar parabola figure.

In the case of a "correspondence", where to a point $x$ in the domain $X$ we assign a set of values $F(x) \subseteq Y$, the natural generalisation of the previous is to allow all pairs $(x,y)$ so that $y \in F(x)$. For a single valued $f$ the set $\Gamma(f)$ is just $\delta(F)$ for the correspondence $x \to \{f(x)\}$, say.

It's like drawing a vertical set for each $x$ with its values on the $y$-axis and taking all these sets together. It could look like a "thickened" normal graph, e.g. but need not. It could be a parabola again e.g. for $x \to \{\sqrt{|x|}, -\sqrt{|x|}\}$ as a correspondence.