0
$\begingroup$

Let $X$ be some topological space and $Y$ some subspace of $X$. I am trying to understand the relation between limits points of a set in the subspace of $Y$ and $X$'s topology. Initially, I thought that $cl_Y(A) \subseteq cl_X(A)$, where $A \subseteq Y$. But it seems that I can prove the two sets are equal; surely this is wrong, but I cannot identify the error in my argument.

First I will prove $cl_Y(A) \subseteq cl_X(A)$. Let $a \in cl_Y(A)$ and $\mathcal{O}$ some open set in $X$ containing $a$. Then $\mathcal{O} \cap Y$ is some nonempty open set in $Y$ containing $a$, implying that $\mathcal{O} \cap Y$ and $A$ intersect. But $\mathcal{O} \cap Y \cap A = \mathcal{O} \cap A$ is not empty, indicating that $a \in cl_X(A)$.

Now, suppose that $a \in cl_X(A)$ and let $U$ be some open set in $Y$ containing $a$. Then there exists some open set $\mathcal{O}$ in $X$ such that $U = \mathcal{O} \cap Y$, which of course implies that $\mathcal{O}$ contains $a$. Hence, $\mathcal{O} \cap A = \mathcal{O} \cap Y \cap A = U \cap Y$ is not empty, proving that $a \in cl_Y(A)$.

So, where did I go wrong?

1 Answers 1

1

When you say $a\in cl_Y(A)$, you automatically assume $a\in Y$

Now if $a\in cl_X(A)$ and $a\in Y$ then $a\in cl_Y(A)$. But there might be points in $cl_X(A)$ which do not belong to $Y$

  • 0
    I think the issue is also found in this sentence: "Now, suppose that $a∈clX(A)$ and let $U$ be some open set in $Y$ containing $a$." I am assuming that there is a open set in $Y$ that contains $a$. – 2017-01-25
  • 1
    Yes, that is the issue indeed. Let $X=\mathbb R$, $Y=A=(0,1)$, $0\in cl_X(A)$, but $0$ does not belong to any open set of $Y$ – 2017-01-25