What keeps Cotton Fabric from falling apart is accomplished in two distinctive different fashions but the physical force is still friction between the different fibers.
Natural cotton fibers are formed in a curly fashion by nature. They are intertwined together naturally and will not simply fall apart. Some of the curls each fiber possess are taken out when the cotton is spun into yarn or thread. Each time this thread is washed, the curls return. This is one of the reasons cotton is stronger when wet than it is when dry.
In the processing of raw cotton, there is a separation by the length of the fibers. It is the longer fibers that are spun into threads and yarn. This is done by first blowing them with a fan and collecting them. In this fashion the ends of the fibers are not uniformed and will naturally get entangled with other fibers. Then the cotton is gathered and spun into yarn or thread. The second method is more modern. It involves the use of needles and an electrostatic charge to get the fibers to bind together.
Immaterial of how the cotton is bonded together there is no product made of a single strand of cotton. Even the smallest thread will have at least two cotton stands twisted together. But it is more common to have three or more. Just like rope, each strand that is twisted together adds to the strength of the thread. When two or more threads are twisted together they are twisted in the opposite direction from the way the threads were originally twisted. This again helps to bind the cotton fibers to each other and forms a longer lasting bond used in today’s textiles and threads used to sew them together. This process is called plying the cotton. After the cotton is made into yarn or thread it can then be weaved tighter. This is a process where the different threads are again intertwined together and pressed against each other.
With all of the twisting in both directions pressing and binding that the cotton fibers are exposed to, it is impossible for anyone or even a machine to pull a single fiber from a textile made of cotton. While some of the finer cotton textiles are gassed, which burns off the lose fiber ends and make the threads smoother, they are still held together with enough strength not to become lose. This process takes the cotton threads and passes them thru an open flame.
This is the cotton that is used for the making of Egyptian sheets and Venetian textiles. When the threads are scorched the dye turns a darker shade but the threads themselves are not burned but still bond tightly together.
The binding of each cotton fiber is such a long and tedious process of twisting and intertwining them, which they will never just fall apart after they are made into a fabric a person would wear. It is just physically impossible.