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I know that the projective plane is represented topologically like this:

enter image description here

But why while applying Van Kampen theorem, it is sometimes drawn like this:

enter image description here

Could anyone explain this to me, please?

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  • $\begingroup$ This is just two different but equivalent ways of drawing a picture of the same space, so maybe which picture an author prefers is merely a personal preference. $\endgroup$ – littleO Oct 4 at 1:38
  • $\begingroup$ @littleO but why they are equivalent? $\endgroup$ – Secretly Oct 4 at 2:19
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    $\begingroup$ A circle and a square are topologically equivalent (that is, homeomorphic), so it seems plausible at least that by identifying opposite points on a square in a certain way, you'd end up with the same space as if you had identified opposite points on a circle in a similar way. $\endgroup$ – littleO Oct 4 at 21:29
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Are you asking why these describe the same space? You can match them up by matching up the two dots in the bottom diagram with, say, the lower left and upper right corners in the top diagram.

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  • $\begingroup$ yes, exactly this is my question $\endgroup$ – Secretly Oct 4 at 2:20

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