Klein bottle contains Möbius band

hbghlyj

A simple closed curve $C$ in a space $X$ is the image of a continuous injection $S^1→X$. Find simple closed curves $C_1, C_2$ and $C_3$ in the Klein bottle $K$ such that

• $K∖C_1$ has one component, which is homeomorphic to an open annulus $S^1×(0,1)$.
• $K∖C_2$ has one component, which is homeomorphic to an open Möbius band.
• $K∖C_3$ has two components, each of which is homeomorphic to an open Möbius band.

[An open Möbius band is the space obtained from $[0,1] \times(0,1)$ by identifying $(0, y)$ with $(1,1-y)$ for each $y \in(0,1)$.]

hbghlyj

If Klein bottle is cut along the dashed curve $C$
(it is closed curve because 4 corners are identified)

$K\setminus C$ has 1 component:

We can shift the right part to the left (as points are identified, this gives the same space):

This is a Möbius band. So part (b) is solved.

hbghlyj

If Klein bottle is cut along the dashed curve $C$
(it is closed curve because 4 corners are identified)

$K\setminus C$ has 1 component:

We can rotate the upper part 180° and stick to the lower (as points are identified, this gives the same space):

This is homeomorphic to an open annulus $S^1×(0,1)$.

hbghlyj

math.stackexchange.com/questions/1039875
can cut the Klein Bottle along the red edge (in reverse, a Möbius band becomes Klein Bottle when the two red edges are identified).

part a) second solution

can cut the Klein Bottle along the blue edge (in reverse, a cylinder becomes Klein Bottle when the two blue edges are identified).

Czhang271828

Here is a short-cut to understand why Klein bottle contains Möbius band:

(1) Klein bottle = Möbius band + quotient relation on its boundary,

(2) There is another Möbius band properly contained in a given Möbius band.

Therefore, Klein bottle contains Möbius band.

hbghlyj

Cutting vertically in the middle

we get 2 Möbius bands

hbghlyj

$C$ is the dashed curve (it is closed curve because 4 corners are identified, and midpoints of opposite sides are identified)

$K∖C$ is the shaded region:

$K∖C$ has only 2 components because the right edge is glued to the left edge:

Each is homeomorphic to a Möbius band.