Are spaces shaped like the digits 0, 8 and 9 homeomorphic topological spaces

general-topologymetric-spaces

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 $\mathbb{R}^{2}$.

Best Answer

$0$ has no cut points.
$8$ has exactly one cut point.
$9$ has infinitely many cutpoints.

To show there are no homeomorphisms among $0,8,9$ use the exercise.

Exercise. Prove if $f:X\to Y$ is homeomorphism and $p$ cutpoint of $X$, then $f(p)$ is cutpoint of $Y$. Also show an arc is not homeomorphic to a circle.

Related Question