I am trying to determine whether the rose with two petals ($S^1 \vee S^1$ or the figure-eight) has a continuous multiplication with identity element. I know that this is true for the unit circle $S^1$ in the complex plane, where $S^1 = \{ z \in \mathbb{C} \mid |{z}| = 1 \}$.
I also know that $S^1$ is a Lie Group, and I believe that because of the intersection point of the figure eight, this space does not have a continuous multiplication with identity element and is therefore not a topological group. Would someone mind pointing me in the right direction for how to approach this idea?