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.


  • $\begingroup$ The initial claim is wrong. If $A=Y$ then $A$ is clopen in $Y$, but can have whatever status (closed, open, both, neither) in $X$. $\endgroup$ – Henning Makholm Nov 10 '15 at 1:12
  • $\begingroup$ 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. $\endgroup$ – Mark LaPolla Nov 10 '15 at 16:47

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 $X = \mathbb{R}$.

  • $\begingroup$ 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. $\endgroup$ – Mark LaPolla Nov 10 '15 at 16:45

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.