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Let $X$ be the space obtained by gluing the diametrical points on the equatorial circle of $S^2$, then how to image $X$.

I calculate the homology group of $X$ and get $H_2(X)=H_0(X)=\mathbb Z,H_1(X)=\mathbb Z_2,$. It implies that $X$ is orientable but is not a closed surface. Can we have a more intuitive description of $X$? For example, what is the boundary of X? How can we glue two $\mathbb P^2$ to get $X$?

Mjr
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    It seems that you're talking about the space mentioned in this post. Also please specify what do you mean by "an intuitive description"? Cell-decomposition or what? – Kevin.S Sep 26 '21 at 08:53
  • Thanks. It seems that we can get $X$ by gluing two $\mathbb P^2$, how can we do it explicitly. And what is the boundary of $X$. – Mjr Sep 26 '21 at 09:02

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