Can my cat have gotten tapeworms from another cat?

Hi all,
I have recently started finding small white sesame things on the bed, where my kitty sleeps during the day. I have made a few internet searches and it appears that he could have tapeworms???
I did not see any signs of fleas on him, and I did put a strong anti parasite medication on him that my vet had given him. He is a strictly indoors cat, he abhors nature, and only travels between my house and my boyfriend's. He does not catch mice. How on earth can he have got them??? The only thing I know is that we briefly cat sitted for a friend whose cat has worms (I do not know why she is not treating the cat), and so he may have used the same litter she used (the cats did not actually meet). Could that have been the reason why he got worms? If so, what precautions should I take in the future? He loves meeting other cats, but I need to make sure he does so safely.

Thank you!

Hi fetching milk Cap.

Getting tapeworm unless your kitty is licking another cat's behind(unlikely) ingesting a flea is how you get it.One flea which will be long gone in the gut.Fleas have tape worm in them.Fleas are in the grass & highly likely in your home.One flea lays over 200,000 eggs in it's life.But ingesting one is all it takes.Or grooming another cat & it has fleas & she ingests it.Swallows it.Very common.HTH C.

How is it possible that my cat has no fleas on him, and he is not scratching, and what do I need to do in order to get rid of the fleas from my apartment?