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A finite set is a set that has numbers you can count. Its not like infinite with no end it has an end.
Some finite numbers in a set: the number of digits on your hand, the number of seats on a bus, and the number of people on earth. Some infinite numbers in a set: the number of positive integers and the number of digits in pi.
They are sets with a finite number of elements. For example the days of the week, or the 12 months of the year. Modular arithmetic is based on finite sets.
This is called a discrete set (all points isolated) or a finite set. Finite sets are always discrete.
Closed sets and open sets, or finite and infinite sets.