In 3 dimensions, the sieve/strainer/colander is an object that's connected with itself despite having holes. There's material that goes around the holes.
In 2 dimensions, the membrane is essentially a line (or a linear sequence of objects) - if that line is interrupted in one space to let stuff through, then it's not connected anymore, it's two separate lines with nothing linking them, it's broken with no possibility to connect "around the hole" because any connection around the hole requires the third dimension, and any connection through the hole closes that hole.
In 2 dimensions, the membrane is essentially a line (or a linear sequence of objects) - if that line is interrupted in one space to let stuff through, then it's not connected anymore, it's two separate lines with nothing linking them, it's broken with no possibility to connect "around the hole" because any connection around the hole requires the third dimension, and any connection through the hole closes that hole.