$\mathbb{R}P^2$ and its lines

Question

I have been solving some past exam questions and I came across the following question. Let $r$ and $s$ two distinct lines in $\mathbb{R}P^2$, and let $X$ the space obtained contracting $r \cup s$ to a point in $\mathbb{R}P^2$. Say if $X$ is a topological manifold. I can not imagine what $X$ is, and i think this is my problem. Help would greatly be appreciated.

Answer 1

Contracting one line to a point gives a sphere. (Why?) Contracting two lines then is equivalent to contracting one loop on a sphere to a point. What do you think you'll get?

Answer 2

The result is the wedge sum of two copy of $S^2$, thus it's not a topological manifold for obvious problems in the pasting point.