Definition of the punctured $\mathbb{R}P^3$

Question

I faced "punctured $\mathbb{R}P^3$" denoted by $\mathbb{R}P^3-\{{pt}\}$ in my studies. I dont know its definition and also my searches in the web are failed. Can anyone help me? What is the visualization of $\mathbb{R}P^3-\{{pt}\}$?

Sorry if my question is trivial!

Thanks

Answer 1

As $\mathbb{R}P^3$ is a manifold, it is inparticular homoegenous. This means that if we remove some point $x\in\mathbb{R}P^3$, it is homeomorphic to if we instead removed some other point $y\in\mathbb{R}P^3$. In that sense, the space $\mathbb{R}P^3$ with a single point removed is uniquely defined up to homeomorphism, so we might as well just talk about 'punctured $\mathbb{R}P^3$' and not worry about which point we're removing.