Topology: An Inquiry-Based Approach

Suppose we try to define a set to be a collection of elements. So, by definition, the elements are the objects contained in the set. We use the symbol $\in$ to denote that an object is an element of a set. So $\notin$ means an object is not in the set — if $x$ is an object in a set $S$ we write $x\in S\text{,}$ and if $x$ is not an object in a set $S$ we write $x\notin S\text{.}$ We write sets using set brackets. For example, the set $\{a,b,c\}$ is the set whose elements are the symbols $a\text{,}$ $b\text{,}$ and $c\text{.}$ We can also include in the set notation conditions on elements of the set. For example, $\{x\in \mathbb{R}:x>0\}$ is the set of positive real numbers. We typically use capital letters to denote sets. Some familiar examples of sets are $\mathbb{R}\text{,}$ the set of real numbers; $\mathbb{Q}\text{,}$ the set of rational numbers; and $\mathbb{Z}\text{,}$ the set of integers. Sets can also contain sets as elements. For example, the power set of a set $S$ is the set of subsets of $S\text{.}$ So the power set of $S=\{1,2\}$ is the set $\{\mathrm{\varnothing },\{1\},\{2\},\{1,2\}\}\text{.}$ (We will define subsets and the empty set later in this activity).