I'm currently trying to verify that the subspace topology is indeed a topology as defined in this post (Verifying the subspace topology with a topology defined in terms of neighborhoods). My problem is with the verification of Axiom c).
Let X denote the space and Y the subspace.
If I assume that $U \subset Y$ is a neighborhood of y with respect to Y then by definition of the subspace topology there is a neighborhood K of y with respect to X such that $U=Y\cap K$.
Suppose that there is $\ L \subset Y$ such that $U \subset L.$
It is obvious that $L=Y\cap L$. Now I would like to prove that $U=Y\cap K $ is a neighborhood of y with respect to X and then use the fact that X is a topological space (for which the axioms a)-d) hold) to conclude that L is a neighborhood of y with respect to X. This would allow me to deduce that L is a neighborhood of y with respect to Y and finish the proof.
But I do not see how to prove that U is a neighborhood of y with respect to X at all. Is this even possible? Or is there some other approach?