# A note on basic set theory

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 [2].

Let $A$ and $B$ be sets. The cardinality of the set $A$ is denoted by $\lvert A \lvert$. A function $f:A \rightarrow B$ is said to be one-to-one (an injection) if for $x,y \in A$ with $x \ne y$, $f(x) \ne f(y)$. A function $f:A \rightarrow B$ is said to map $A$ onto $B$ (a surjection) if $B=\left\{f(x): x \in A\right\}$, i.e. the range of the function $f$ is $B$. If the function $f:A \rightarrow B$ is both an injection and a surjection, then $f$ is called a bijection, in which case, we say both sets have the same cardinality and we use the notation $\lvert A \lvert = \lvert B \lvert$. When the function $f:A \rightarrow B$, we denote this condition by $\lvert A \lvert \le \lvert B \lvert$. The Cantor–Bernstein–Schroeder theorem states that if $\lvert A \lvert \le \lvert B \lvert$ and $\lvert B \lvert \le \lvert A \lvert$ then $\lvert A \lvert = \lvert B \lvert$ (see 1.12 in [2]).

For the functions $f:X \rightarrow Y$ and $g:Y \rightarrow Z$, we define the function $g \circ f$ by $(g \circ f)(x)=g(f(x))$ for each $x \in X$. The function $g \circ f$ is denoted by $g \circ f:X \rightarrow Z$ and is called the composition of $g$ and $f$.

We use the notation $B^A$ to denote the set of all functions $f:A \rightarrow B$. It follows that if $\lvert A \lvert \le \lvert B \lvert$ then $\lvert A^C \lvert \le \lvert B^C \lvert$. To see this, suppose we have a one-to-one function $f:A \rightarrow B$. We define a one-to-one function $H: A^C \rightarrow B^C$ by $H(h)=f \circ h$ for each $h \in A^C$. Since $f:A \rightarrow B$, it follows that $H$ is one-to-one.

By $\omega$, we mean the first infinite ordinal, which can be viewed as the set of all nonnegative intergers. By $\omega_1$ we mean the first uncountable ordinal. The notation $2^\omega$ has dual use. With $2=\left\{0,1\right\}$, the set $2^\omega$ denotes all functions $f:\omega \rightarrow 2$. It can be shown that $2^\omega$ has the same cardinality as the real line $\mathbb{R}$ and the unit interval $[0,1]$ and the middle third Cantor set (see The Cantor set, I). Thus we also use $2^\omega$ to denote continuum, the cardinality of the real line.

If $\lvert A \lvert=2^\omega$, then the set $\lvert A^{\omega} \lvert=2^\omega$ where $A^{\omega}$ is the set of all functions from $\omega$ into $A$. Since $\omega_1$ is the first uncountable ordinal, we have $\omega_1 \le 2^\omega$. The Continuum Hypothesis states that $\omega_1 = 2^\omega$, i.e. the cardinality of the real line is the first uncountable cardinal number.

The union of $2^\omega$ many sets, each of which has cardinality $2^\omega$, has cardinality $2^\omega$. Furthermore, the union of $\le 2^\omega$ many sets, each of which has cardinality $\le 2^\omega$, has cardinality $\le 2^\omega$.

Reference

1. Kunen, K. Set Theory, An Introduction to Independence Proofs, 1980, Elsevier Science Publishing, New York.
2. Willard, S., General Topology, 1970, Addison-Wesley Publishing Company.