This question actually has two parts:
a. Prove that every subspace of a topological space with the discrete topology has the discrete topology.
b. Prove that every subspace of a topological space with the trivial topology has the trivial topology.
If I can do part a, part b will be easy. What do I need to demonstrate, exactly, to show that a is true?
Best Answer
Suppose we have a topological space $(X, \tau_X)$, where $\tau_X\subseteq 2^X$ is a topology on $X$. Then every subset $Y\subseteq X$ we can consider as a topological subspace of $(X, \tau_X)$ with induced topology $\tau_Y$. This topology constructs simply like this: $$ \tau_Y = \{U\cap Y:\ \ U\in\tau_X \} $$
So (a) if you have a topological space with the discrete topology $(X, 2^X)$ then for every $Y\subseteq X$ you'll have induced topology $\tau_Y = \{U\cap Y:\ \ U\in 2^X \}$ which is $2^Y$, just as was stated.
And (b) if you have a topological space with the antidiscrete (trivial) topology $(X, \{\emptyset, X\})$, then the induced topology on $Y$ will look like this $\tau_Y = \{U\cap Y:\ \ U\in \{\emptyset, X\} \}=\{\emptyset, Y\}$, just as expected.