No, the whole asbestos/cancer thing is blown WAY out of proportion.
Yes, there is a distinct link between inhaling asbestos fibers and respiratory cancers such as mesothelioma. But you have to be exposed to quite a lot of asbestos to have a problem, and it has to be in a fibrous form that your respiratory system cannot easily expel the fibers. People who installed or repaired asbestos insulation and things like that are the ones who must worry.
Unfortunately, greedy lawyers have got everyone ELSE so worked up about it that everyone thinks they are going to die if there is a one inch square of asbestos within a hundred miles of them.
The "popcorn" used in ceiling texturing was made from a small amount of asbestos in a binder (like paint). This binder, and all subsequent paintings of the ceiling, will have completely encapsulated all the asbestos fibers to the point that they cannot be inhaled in a form that is dangerous to you -- ESPECIALLY not from just a one-time exposure. If you were a professional asbestos remover, exposed to it every day for your working life, you MIGHT have a concern. Think of it like this: you know that pink fiberglass insulation material in some attics, and how nasty that stuff is to work with? You know it's not going to be good for you to breathe that stuff in all day long. But cars and boats and houses and all sorts of things have fiberglass parts and bodies, and that doesn't irritate our systems -- because it is encapsulated in a resin and it stays that way. If you WORKED with fiberglass fibers day in and day out you would have to worry about chronic exposure issues. But just casual contact in day to day life will not expose you to enough to worry about.
Think about it -- automobile brake linings were made from asbestos for over 50 years. Practically every car in the world shed asbestos fibers whenever the brakes were applied. And every home in America had those ugly "popcorn" ceilings. And guess what, if your floors had those grayish colored tiles then it was made of asbestos too. Asbestos was used in home siding, pipe insulation, caulks, paints, and just about everything you can think of for decades. If asbestos was all that big of a problem from casual contact, there would not be anyone left alive in this country today.
I would definitely remove that "popcorn" ceiling from your home -- but just because they are ugly as sin and collect dirt and dust worse than a modern textured ceiling. Not because they have asbestos in them. If you are really worried and don't want to go through the hassle of removing the popcorn, ask your landlord if it is OK to paint it over with a layer of paint.
Oh yeah, and if the landlord disclosed the presence of asbestos in the lease, no you can't sue him over it. Even if he hadn't disclosed it, the most you could have gotten from the fact was to be released from your lease obligations.
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