What is the meaning of “the inner leaflet of their lipid bilayer corresponds to the outer leaflet of the plasmalemma.” The whole quote is, “Their lumen corresponds topologically to the outside of the cell; consequently, the inner leaflet of their lipid bilayer corresponds to the outer leaflet of the plasmalemma.” The “they” corresponds fo the vacuolar apparatus of a cell. My question here is, how is it possible for the inner leaflet of the vacuolar apparatus to lie on the outer leaflets of the plasma membrane? Wouldn’t this mean that the vacuolar apparatus partly lies outsides the cell? Or am I mistaken? The excerpt is from Chapter 4 of Principles of Neural Science by Kandel et al.
I don't have Kandel in front of me but I am pretty sure I know what your book is trying to say. The specific language they've chosen for this section is quite odd and I believe old-fashioned, though, which might be leading to your confusion.

Transferring membrane-bound material from intracellular organelles to the surface of a cell involves or vice-versa involves fusing vesicles with the plasma membrane or budding vesicles from the plasma membrane. The process is more or less the same, just in reverse. See the diagram below:

Adapted from https://en.wikipedia.org/wiki/Golgi_apparatus#/media/File:0314_Golgi_Apparatus.jpg
Notice how the &quot;protein&quot; has its purple plus-shaped domain on the outside of the cellular membrane? If it were to be secreted in a vesicle (as pictured here) or taken up by endocytosis (just imagine the arrows pointing the other way), fusing or pinching the membrane would mean that this &quot;outside&quot; part is actually on the inside of the secretory vesicle, Golgi apparatus, any lysosome/endosome, etc. What stays constant is the part facing the cytosol.
Therefore, you can say that the &quot;inside&quot; of any of these compartments &quot;looks like&quot; the outside of the cell, whereas the outside of these compartments (facing the cytosol) looks like the inside of the cell membrane (also facing the cytosol).
For another intuitive way of thinking about this, imagine you have an inflated balloon. You press a marble into the side of the balloon and pinch inside so now the marble is entirely inside this little bubble inside the whole balloon. It's still covered by &quot;outside balloon&quot;, but yet it is inside this little compartment you've created.
For more general thoughts on &quot;topology&quot; (what is referred to by &quot;topologically&quot;) see the general mathematical concept: https://en.wikipedia.org/wiki/Topology