0
$\begingroup$

Let $X$ is a topological space, $Y$ is a subspace of $X$, $A \subseteq Y$. Then I know $$Y\cap Cl_X(A)=Cl_Y(A)$$ holds.

But does $$Y\cap Cl_X(X-A)=Cl_Y(Y-A)$$ also holds?

$\endgroup$
  • $\begingroup$ So your question is $Cl_Y(Y-A)=Cl_Y(X-A)$? $\endgroup$ – Siminore Sep 14 '12 at 15:02
  • $\begingroup$ @Siminore I think not, because $X-A$ may not included in $Y$. $\endgroup$ – Popopo Sep 14 '12 at 15:32
2
$\begingroup$

The assertion is false.

Take for example $X = \mathbb R, Y = [0, 2]$ and $A = [0, 1]$. Then $0 \in Y \cap Cl_X(X - A)$ but $0 \not \in Cl_Y(Y - A)$.

$\endgroup$
  • $\begingroup$ So that means $Int_Y(A)=Y \cap Int_X(A)$ is also not true in general. $\endgroup$ – Popopo Sep 15 '12 at 4:17
  • $\begingroup$ Yes, the same example applies to it. $\endgroup$ – Levon Haykazyan Sep 15 '12 at 13:09

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.