Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered.
Is the space below also a Klein bottle ($d$ and $e$ have no direction)?
I know that if the black lines, $d$ and $e$, were removed then we would have a Klein bottle as it would reduce to this:
But do the black lines stop it from being topologically equivalent to a Klein bottle, and if so, how?
Thank you