Consider the topological spaces shaped like the numerals "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 $\mathbb{R}^{2}$.