>> That might explain it, I guess. So if I see something before me that I
>> am inclined to refer to as a "coffee cup", how am I to tell whether or
>> not it actually _is_ a coffee cup in some essential way?
>
> In fact, it could be your doughnut.
>
> (Old topology joke.)

My doughnuts are generally not continuously deformable to coffee cups.
I'd have to poke them with a stick to make that true, and that would
squirt jam everywhere.