Open subsets of Jet Bundles

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I am currently reading through 'Stable mappings and their singularities' by Golubitsky and Guillemin. There they define $$ J^k(X, Y)_{x, y} $$ to be the set of equivalence classes of smooth maps $f: X\to Y$ with $f(x) = y$ under the equivalence $\sim_k$, where $f \sim_k g$ if $f, g$ have $k$-th order contact at $x$. Then they define the Jet Bundle as $$ J^k(X, Y) := \sqcup_{\{x, y\} \in X\times Y} J^k(X, Y)_{x, y} $$

Later they go on to define charts on $J^k(X, Y)$: given $(\phi, U), (\psi, V)$ charts on $X, Y$ respectively, they define a chart $$ J^k(U, V)\to \phi(U) \times \psi(V) \times B_{n, m}^k $$ where $B_{n, m}^k$ is a space consisting of $m$-tuples of polynomials in $n$ variables up to and including degree $k$, with zero constant term. What bothers me is related to the fact that they do not even define a topology on $J^k(X, Y)$. Implicit in their definition of the above charts is the assumption that $J^k(U, V)$ is even a subset of $J^k(X, Y)$, clearly it isn't strictly a subset with the above definitions, but what would a natural injection look like? I know that jets carry local information, so there should be a natural injection, but I can't see one with these definitions.

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From what I can tell they are defining the topology on $J^k(X,Y)$ by the given charts. $J^k(U,V)$ is a subset of $J^k(X,Y)$, bearing in mind that in the jet setting we are viewing functions purely as the coordinates of the jet space. i.e. you don't have to worry about the difference between $f|_U$ and $f$, they both have the same coordinates (value of $f$ at its $k$ derivatives at the point).