0
$\begingroup$

I have a proof where it says the pre-images of an open set is open, but the open set is like this: (X ∩ Y) is open, where X is open and Y is not open. Is the statement true? How to prove it?

  • 1
    Is what statement true? – 2017-01-30
  • 0
    (X ∩ Y) is open, where X is open and Y is not open – 2017-01-30
  • 1
    Take $Y=[-10,10]$ and $x=(-1,1)$. There is no contradiction. – 2017-01-30
  • 1
    Google 'relative topology'. – 2017-01-30
  • 0
    and how could I prove it if I haven't take a course of Topology?, just Mathematical Analysis – 2017-01-30
  • 1
    @copper.hat: ... and then take $X=(−10,10)$ and $Y=[−1,1]$ – 2017-01-30

1 Answers 1

1

The theorem is false: notice $\mathbb R^2$ is open but a straight line $l$ is not open. Notice that $\mathbb R^2 \cap l$ is not open.

  • 0
    It is open in the subspace topology of the line, which is probably the context the theorem was used. The question is a bit lacking in that regard. – 2017-01-30
  • 1
    Hmm Good point, although in that case no proof would be requiere. It is by definition. – 2017-01-30