# Clarification about the subspace topology

Let $(X,\tau )$ be a topological space and let $A,B \subseteq X$ be subspaces of $X$ with their respective subspace topology inherited from $X$. consider the set $A \cap B$.

I'm not quite sure if the following topologies on $A \cap B$ are all the same:

i) the subspace topology on $A \cap B$ inherited from $X$

ii) the subspace topology on $A \cap B$ inherited from $B$

iii) the subspace topology on $A \cap B$ inherited from $A$

Let $U$ be an open set in $A\cap B$ in the topology inherited from $X$. Then there is an open set $V$ in $X$ such that $V\cap A\cap B=U$, so $U$ is an open set in the topology inhertited from $A$ and from $B$.
Now consider an open set $U$ in $A\cap B$ in the topology inherited from $A$. Then there is an open set $U'$ in $A$ such that $U=U'\cap(A\cap B)=U'\cap B$. But there is also an open set $V$ in $X$ such that $U'=A\cap V$. Then $U=A\cap V\cap B$. Then $U$ is an open set in the topology inherited from $X$.
Suppose $U$ is an open subset of $A\cap B$ in the subspace topology inherited from $A$. Then there is an open subset $V$ of $A$ in the subspace topology inherited from $X$ such that $V\cap (A\cap B)=U$. Also, there is an open subset $W$ of $X$ such that $W\cap A=V$. But $W\cap (A\cap B)=(W\cap A)\cap (W\cap B)=V\cap(W\cap B)=V\cap A\cap B$. So, you can see that that the subspace topology $A\cap B$ inherits from $A$ is the same as the topology it inherits from $X$.