Lee has as an exercise that the subspace topology is the unique topology that satisfies the characteristic property which is for $S\subset X$ a function $F:Y \to S$ is continuous iff the composition $i_S\circ F$ is continuous.
I think that he means that the only topology that this is true is the subspace topology, but I’m not sure how to prove that no other topology satisfies the property. My first question is what needs to be shown in a proof to say that it is the unique topology, and once I know this I should be able to apply it to the exercise.
Best Answer
Suppose that $\tau$ is any topology on $S$ that satisfies the "characteristic property".
Then let $1_S$ be the identity on $(S, \tau)$, which is always continuous (for any topology on $S$) so we know that $i_S \circ 1_S = i_S$ is continuous (using the "characteristic property" in one direction). This implies that if $U$ is open in $X$ then $i_S^{-1}[U] = U \cap S$ is open in $\tau$. So the subspace topology $\tau_S$ obeys $\tau_S \subseteq \tau$.
Define the function $f(x)=x$ from $(S,\tau_S)$ to $(S,\tau)$, and note that $i_S = i_S \circ f$ as a function $ (S, \tau_S) \to X$ is continuous (we know the subspace topology $\tau_S$ makes this map continuous) and so $f$ is continuous (by the other direction of the "characteristic property"), which implies that for all $U \in \tau$, $f^{-1}[U]=U \in \tau_S$ or $\tau \subseteq \tau_S$ and so $\tau = \tau_S$ and we have unicity of $\tau_S$.