I have to show hat the following two figures are homeomorphic to each other. Let the first and second figures be denoted as $X,Y$ respectively.
$X$ is is obtained by sewing together three twisted strips of paper to two circular discs of paper and $Y$ is obtained by sewing together two long strips of paper as shown in the figure.
My attempt: Since a double twisted Mobius band is homeomorphic to a cylinder, figure $X$ is homeomorphic to "A Mobius band with a small disk removed from it's interior" ( I hope it is fine. If not, you can give your argument why is it not right and then proceed in another way.) Also we can show $Y$ is homeomorphic to "A torus with a small disk removed from it". Now we know that if $f:Y\rightarrow X$ is a homeomorphism then $f(\partial Y)=\partial X$. Obviously $f|_{\partial Y}$ is continuous. But here as per my description $\partial X$ has two components while $\partial Y$ has only one component, hence they can't be homeomorphic to each other.
So please find the fault in my argument and give me a hint for proving this! If these are not homeomorphic, then also give me some hint for disproving this(though I've disproved it, i need to hear if there is any other way we can think about this).
Also note that, both have same homotopy type as both of them deform retract to figure $\infty$ , so there is a chance of these figures to be homeomorphic.
P.S. Well, you can provide me an intuitive answer also!
I don't think the figure $X$ is a Mobius band with a disc removed. For one thing, it has only one boundary component (just by tracing your finger around the arcs and seeing that cover everything). Secondly, it is orientable, since you always pass through an even number of twisted bands when you travel by any closed curve.
If you're able to use the classification of surfaces theorem, then you could verify the two are homeomorphic by calculating the Euler characteristic (which is $-1$ for both), number of boundary components ($1$ for both), and the fact both are orientable.
For a more direct way to see they are homeomorphic, we start by looking at the figure $Y$. It is an annulus with a strip glued onto it. So we want to describe $X$ in the same way. Now if we cut one of the twists from the figure $X$, we are left with a band with two twists in it, which is homeomorphic to an annulus. So $X$ is in fact homeomorphic to an annulus with a band glued to it. There are two ways to glue on a band, one which results in an orientable surface and one which results in a non-orientable surface, and we know $X$ is orientable. So the surface $X$ is homeomorphic to the orientable surface where we glue a band to an annulus, which is also homeomorphic to $Y$.