Closed subschemes as subsets A closed subscheme $Z$ of a scheme $X$ is by definition an equivalence class of closed immersions. How can one view $Z$ as a subset of $X$ (on the level of topological spaces)? Can one define a closed subscheme structure on an arbitrary closed subset $Y$ of $X$?
 A: 0) Hartshorne's definition of closed subscheme, which you use, is surprisingly bad for a mathematician of his calibre.
(His definition of open subscheme is weird too: see  here).
The correct definition, as given by Grothendieck, Mumford, Qing Liu, Görtz-Wedhorn, De Jong's Stacks Project, etc. is the following:  
1)  A closed subscheme  of the scheme  $(X,\mathcal O_X)$ is a scheme $(Z,\mathcal O_Z)$ such that $Z\subset X$ is a closed subspace (with inclusion denoted $j:Z\hookrightarrow X$)  and such that the sheaf of local rings $\mathcal O_Z$ is the restriction to $Z$ of the sheaf $\mathcal O_X/\mathcal I$, where $\mathcal I$ is some quasi-coherent ideal sheaf $\mathcal I\subset \mathcal O_X$.
Such a closed subscheme comes equipped with an obvious closed immersion $(Z,\mathcal O_Z)\hookrightarrow (X,\mathcal O_X)$.  
2) a) Given an arbitrary  closed subset $Z\subset X$ of a scheme $(X,\mathcal O_X)$ there exist in general infinitely many closed subschemes $(Z,\mathcal O_Z)\subset  (X,\mathcal O_X)$.
For example the affine line $X=\mathbb A_k^1$ has as closed subschemes with support the   origin $\{0\}\subset X$ all the subschemes with structure sheaf $\mathcal O_X/\mathcal I^n$, where $\mathcal I$ is the sheaf of sections vanishing at $0$.
Of course the closed subscheme $(\{0\},\mathcal O_X/\mathcal I^n)$ is isomorphic to $\operatorname {Spec}( k[T]/(T^n))$.   
b) However among the multitude of closed subschemes with $Z$ as supporting topological space there is a privileged one: the unique closed subscheme with reduced structural sheaf $\mathcal O_Z$, corresponding to the largest quasi-coherent ideal sheaf $\mathcal I\subset \mathcal O_X$ such that $\mathcal O_X/\mathcal I$ has support $Z$.
In the degenerate case where $Z=X$, that privileged closed subscheme is a closed subscheme called the reduced subscheme $X_{red}\subset X$ associated to $X$.
