# How can I visualize ${A^0}^{Y} \supset A^0$?

$$(X,\tau)$$ is a topological space and $$Y\subset X$$.How can I visualize $${A^0}^{Y} \supset A^0$$,where $$A\subset Y$$.The proof is not very difficult but the fact is not seeming very obvious to me.Suppose $$x \in A^0$$ then $$\exists U_x$$ open such that $$x\in U_x \subset A$$.Taking intersection with $$Y$$,we get $$x\in U_x\cap Y \subset A\cap Y$$.$$U_x\cap Y$$ is open in $$Y$$ as $$U_x$$ is open in $$x$$ and in fact $$U_x \cap Y=U_x$$ and $$A\cap Y=A$$ because $$A\subset Y$$.So,$$x\in {A^0}^Y$$.But how to feel that this should be true,I am talking of an intuitive feeling that should be obvious even without calculation.

• What is $A^0$ and $A^{0^Y}$? – supinf Mar 9 at 11:36

You can look at it this way: $$A^\circ$$ is the largest open (from point of view of $$X$$) subset of $$A$$, $$A^{\circ^Y}$$ is the largset open (now form point of view of $$Y$$) subset of $$A$$. Clearly, every $$X$$-open subset of $$Y$$ is $$Y$$-open. An these facts together are enough.
• @KishalaySarkar The diagram shows that it may happen that the $Y$-interior is strictly larger than the $X$-interior. – user87690 Mar 9 at 11:52
• It is true that $\partial_Y A\subset \partial A$. – Kishalay Sarkar Mar 9 at 12:03