What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$? When dealing with a space $X$, that posses a lot of structure (complete lattice, complete metric space, vector space), what can be said about the cartesian exponentiation $X^Y=\{f \mid f:Y\rightarrow X\}$ (whic is the set of functions from another set $Y$ to $X$)?
It is intuitivly clear, that the point-wise versions of operations and relations in $X$ carry over many properties of $X$ to $X^Y$. (e.g. $X^Y$ is also a vector space  with point-wise addition and multiplication, it is also a complete lattice with the point-wise or product order. It is also a complete metric space under the maximum (or point-wise) metric).
Is there a field of mathematics, that is concerned with this particular sharing of structure between a space $X$ and its function space $X^Y$?
Or more pragmatically, what results can I use, in order to spare me the proof of these 16 mostly trivial and shallow implications (8 for vector space, 3 for partial order + 1 completeness, 3 for metric + 1 completemess)?

The slice category $\mathbf{C}/C$ is the category of arrows with codomain $C$  and "commutative triangles" between them. In $\mathbf{Set}$, one can think to $\mathbf{Set}/X$ as the category of $X$-valued functions. 
A thoerem of Freyd (Theorem 2.1 of Toposes, Triples and Theories) states that the property of $\mathbf{C}$ being a topos (which $\mathbf{Set}$ enjoys) is inherited by $\mathbf{C}/C$. So the slice category has a lot of structure, notably finite (co-)completeness, existence of exponentials, suboject classifier as well as the Heyting lattice structure of the subobjects of the terminal object. For a richly exemplified book on topoi, see Goldblatt's book.
If your space of interest is a topos, using the previous result may translates some of the properties you are interested in.
