Topology: General Topology 2021

Section 5.1 Sets and elements

The first thing you have to know about sets is that a set \(X\) is equal to a set \(Y\) if and only if \(X\) and \(Y\) have the same elements. That is, \(X=Y\) if and only if for every \(A\text{,}\) one has \(A\in X\) if and only if \(A\in Y\text{.}\)

The easiest set in the world is the empty set \(\varnothing\text{.}\) It has no elements. The sentence \(x\in\varnothing\) is always false, no matter what \(x\) is. That means that any universally quantified sentence over the empty set — i.e., \((\forall x\in\varnothing)(\phi(x))\) — is true, and any existentially quantified sentence over the empty set — i.e., \((\exists x\in\varnothing)(\phi(x))\) — is false.

The next easiest sets in the world are singletons. A singleton is a set with exactly one element, \(\left\{X\right\}\text{.}\) Don't forget that that \(X\) has to be a set. The axioms of set theory let you take any set \(X\) and build the singleton \(\{X\}\text{.}\) More generally, if we have a pair of sets \(X,Y\text{,}\) we're permitted to form the set \(\left\{X,Y\right\}\text{.}\)