0
$\begingroup$

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 $A$

$\endgroup$
2
$\begingroup$

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 $X$.

$\endgroup$
0
$\begingroup$

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 $X$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.