If A is closed in subspace Y when is A open in the larger space X?

Y is a subspace of X. If A is clopen in Y and Y is open in X then A is open in X. However, if A is closed in Y then A is closed in X. Correct?

Is there a case where A is closed in Y but open in X, no matter whether Y is open or closed or both in X?

If we consider the following subset of the real line

Y= [0,1] U (2,3)

in the subspace topology. The set [0,1] is open since it is the intersection o the open set (-1/2,3/2) of R with Y. Similarly, (2,3) is open as a subset of Y; it is open as a subset of R. Since [0,1] and (2,3) are complements in Y of each other, they are closed as subsets in Y.

Now, I need an example where A is closed in Y but open in the larger space of X.

Thanks.

• The initial claim is wrong. If $A=Y$ then $A$ is clopen in $Y$, but can have whatever status (closed, open, both, neither) in $X$. – Henning Makholm Nov 10 '15 at 1:12
• Yes. I should have been clearer. I meant that one of the choices was open. I was wondering how to get A to be simply open and not clopen in X. Thanks. – Mark LaPolla Nov 10 '15 at 16:47

1 Answer

The subset $A = (2,3)$ is closed as a subset of $Y$ (it is the complement of $[0,1]$ as you stated), but open in $X = \mathbb{R}$.

• Ah, I was thinking that I had to have a an X that also had [0,1] open in X. I should have seen this myself. Easy. Thanks again. – Mark LaPolla Nov 10 '15 at 16:45