This is a short note listing some basic facts on set theory and set theory notations, mostly about cardinality of sets. The discussion in this note is useful for proving theorems in topology and in many other areas. For more information on basic set theory, see .
Let and be sets. The cardinality of the set is denoted by . A function is said to be one-to-one (an injection) if for with , . A function is said to map onto (a surjection) if , i.e. the range of the function is . If the function is both an injection and a surjection, then is called a bijection, in which case, we say both sets have the same cardinality and we use the notation . When the function , we denote this condition by . The Cantor–Bernstein–Schroeder theorem states that if and then (see 1.12 in ).
For the functions and , we define the function by for each . The function is denoted by and is called the composition of and .
We use the notation to denote the set of all functions . It follows that if then . To see this, suppose we have a one-to-one function . We define a one-to-one function by for each . Since , it follows that is one-to-one.
By , we mean the first infinite ordinal, which can be viewed as the set of all nonnegative intergers. By we mean the first uncountable ordinal. The notation has dual use. With , the set denotes all functions . It can be shown that has the same cardinality as the real line and the unit interval and the middle third Cantor set (see The Cantor set, I). Thus we also use to denote continuum, the cardinality of the real line.
If , then the set where is the set of all functions from into . Since is the first uncountable ordinal, we have . The Continuum Hypothesis states that , i.e. the cardinality of the real line is the first uncountable cardinal number.
The union of many sets, each of which has cardinality , has cardinality . Furthermore, the union of many sets, each of which has cardinality , has cardinality .
- Kunen, K. Set Theory, An Introduction to Independence Proofs, 1980, Elsevier Science Publishing, New York.
- Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.