# Tagged Questions

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### How can I prove formally that the projective space is a Hausdorff space?

I want to prove the Hausdorff property of the projective space with this definition: Define $\mathbb{P}^n$, the real projective space of dimension n to be the set of 1-dimensional linear subspaces ...
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### Definition of real projective line

On $\mathbb R^2-(0,0)$ we define the following equivalence relation: Two points $(x,y)$ and $(x_0,y_0)$ in $\mathbb R^2-(0,0)$ are equivalent if there exists $a\in \mathbb R^*$ such that x=ax_0,\; ...
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### Cutting the $2$-dimensional real Projective Space

I have the following question. Let $M$ be a smooth manifold which is homeomorphic to $\mathbb{R}P^{2}$. If one cuts $M$ along a non-contractible path then $M$ should be homeomorphic to a closed disc, ...
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### Topology of the Segre product vs. the product topology

In general, the product topology on two (quasiprojective) varieties is not the same as the topology of the product variety given by the Segre embedding. This is something I've often seen asserted is ...
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### details and reference (for undergraduate student) on the constructions and topology of $\mathbb{RP}^n$

I want to know if there is a "natural" topology on $\mathbb{RP}^n$, if yes, how it is defined? (a natural topology for me is a topology which, unless said differently, it is considered without said ...
Here's my question. Let's consider the polynomial $p(y_{1},...,y_{n})$ with $deg(p)=d$. The set $C=\{(y_{1},...,y_{1})\in \mathbb{R}^{n} | p(y_{1},...,y_{n})=0\}$ is closed in $\mathbb{R}^{n}$ ...