Consider the topological spaces “0”,”8″ and “9” in $ \mathbb{R}^{2}$ . Are they homeomorphic?
I have an approach that doesnt look very rigorous to me. I wanted to know how to formalize this if its correct.

0 and 8 are not homeomorphic since excluding one point of 0 the space is still connected, but excluding the “tangent point” of 8, we have a disconnected space.

Same idea for 8 and 9.

The space 9 is union of one circle and one arc. The arc is homeomorphic to the circle, so we can view 9 as a union of two circles, then 8 and 9 are homeomorphic
PS: the topology of the spaces is induced by topology of $ \mathbb{R}^{2}$ .